The double Mellin-Barnes type integrals and their applications to convolution theory

V >t

= "p

l

Ff

r[a.-a.s-A.t] +i

j

j

77

j

m

r[l-b.+0.s+B.t] j

j=i

j

J

n

rf

r ( d -6 s)

(1.16)

* (s) = - ^

rf

—

^2

77 j=m

,

q

*i j=n

J

j

>—

ni-c.+r s) 1

r(l-d.+5.s) 7 7 +i

J

J

j=m

-^—

2

,

r(c.-y.s) +i

J

J

(e ,E ) , j j I,P3

(f ,F ) and J J i,q3

all

a ,A , /3 ,B ,5 ,r J J j j j J

are

real

positive

numbers.

The

standard

work

on

the

theory

and

applications

of

the

last

H-function (1.14) of two variables is the monograph by H.M.Srivastava, K.C.Gupta and S.P.Goyal (1982) where an extensive bibliography of all relevant papers up to 1982 is also contained.

New trend in the theory of hypergeometric functions was initiated by R.G. Buschman who in 1978 defined the most general H-function of two variables as follows

H[x,y; (a,a,A) ; (j3,b,B) ; L g ,L ] (1.17) 9(s, t)x s y~ dsdt, (2TTI) 2

L

s

L. t

where T~Tr(a +a s+A t) 1 _'

(1.18)

j

j

j

e(s,t) = - ^ n "TTr(/3 +b s+B t) J=I

Here coefficients a ,A , b ,B are real J J J J positive, negative or equal to zero.

numbers and they y

may be y

R.G. Buschman has considered various aspects of the last H-function in (1.17) in his several papers (1977-1990).In particular, in the study of convergence of the integral of (1.17) he has found some inaccuracies which were made by other authors. This mistake means that the double Mellin-Barnes type contour integrals in (1.8), (1.11) and (1.14) may be divergent, but at the same time their both inside integrals converge. In

the present

chapter

(§3-4) we

shall

consider

in detail

convergence problem and give its complete solution established

8

the

recently

by the first author. The paragraphs §5 — §9 are devoted to the study of various properties of the Buschman H-function (1.17) in general case.

§2. Definition and notation

Note that Fox's H-function (1.5) can be written in the alternative form as follows

T T r ( a +a s) (2.1)

H[x;(a,a)

1

; 0 , b ) ;L ] m n s

J= i

x

27Ti

ds,

L T T f O / b s) S

where a , b j

It

J= l

are only real, positive or negative numbers. j

allows

us

to generalize this function in the case

of two

variables in the following way.

Definition 2.1. General H-function of two complex variables x and y is called the convergent double Mellin-Barnes integral H[x,y; (a,a,A) ; (0,b,B) ; L ,LJ J m n s t (2.2) 1

9(s,t)x s y

dsdt,

(27Ti)2 . I

- t 1 Js

where

"TTr(a + a (2.3)

e(s,t)

S+A

t)

j=i

TTr(/3 +b s+B t) .

J=I

9

j

j

J

Here m and n are non-negative integers (an empty product is inter­ preted

as unity);

a., b., A., B. are

a 2 +A 2 * 0, b 2 +B * 0: a , 6 are J J J J J J

real

complex

numbers

such

that

numbers. The variables x and

y are not equal to zero and x

s

= exp{-s[log|x| + iarg(x)]}

and y

= exp{-t[log|y| + iarg(y)]},

in which log|x| and loglyl denote the natural logarithms of |x| and |y|. Also,

the L

and

s

L. are t

infinite

contours

in s-plane and

t-plane,

respectively, such that a. + a s + A.t * 0,-1,-2, ...for s e L , t e L , j = 1,2,...,m. Here suppose that L , L,., a , a , A , 8 , b , B , x and y satisfy some **

s'

t'

j'

j' j

j

j'

j

reciprocal conditions which provide the convergence of the integral in (2.2). These

conditions are

discussed in

detail

in

the

following

paragraphs §3 — §4. Remark 2.1. The H-function contains defined

(2.2) is

very a general object and it

all other G- and H- functions of two variables which were by

various

authors

in

literature

(see,

for

example,

(1.8)

(1.11), (1.14)).

Remark 2.2. The H-function

(2.2) is different from the H-function

which was defined by R.G.Buschman (1978) x

by x and y

by y. It means that x s y

only

by the replacement of

is replaced by xsy

in the

integral (2.2).

We make this replacement in connection with its large convenience in the further

studies and

applications

of

the H-function

(2.2).

This

question is connected with the double Mellin transform of the H-function and it will discussed in detail in §7.

Remark 2.3. Here, as in R.G.Buschman (1978), in

Definition 2.1 we

omit the poles separation supposition for the kernel 9(s,t) (2.3) in the

10

integral

(2.2).

(respectively

This

L.)

known

must

supposition

separate

the

means

poles

that

of

1

the

1

T

contour

L

r (a +a s+A t) j

a >o

j

j

j

(respectively

m ] |" T(a>+a.s+A t)) from J J J A >o

m the poles of ] |" T(a +a s+A t) J j j a
J

j

m (respectively

] f" r(a.+a.s+A.t) under t € L J J J A
(respectively s € L ). s

j

Here

m ] f" a >o

denotes

the

product

over

all

j = l,...,m,

for

which

j

a

> 0, and so on. j

In §3 and §4 we shall prove that this supposition does not influence the convergence of the integral in (2.2).

Till

1977

the H-function

of

the

kind

(2.2) was

introduced

and

studied by R.G.Buschman (1977-1990). He suggested many interesting ideas which are reflected and used in this Chapter. Later in 1983 O.l.Marichev and Vu Kim Tuan discussed various types and notations, applicable to the study of analogous H-function of N variables. By nowadays a great number of original papers of various authors are dedicated to the study of the H-function

(2.2) in general or special

cases. In particular, the theory and applications of H-function

(1.14)

are considered in the book by H.M.Srivastava, K.C.Gupta and S.P.Goyal (1982).

This Chapter presents new results obtained by the first author in the theory of the general H-function (2.2). In particular, the following §3 and

§4 are devoted

to

the complete

problem of the general H-function.

11

solution

of

the

convergence

§3. The convergence region of the general H-function of two variables Now suppose that the contour L c+ioo. Then integral one

variable

in

in the s-plane runs from c-ioo to

(2.1) which is used to define Fox's H-function of

alternative

form

converges

if

(see

A.L.Dixon

and

W.L.Ferrar (1936)) ( m n ^ \ Y. la.I - E lb I • I

|arg(x)| < -j-

(3.1)

Consequently, if the contour L

in the t-plane also runs from C-ico

to C+ico, then condition (3.1) together with the next inequality (3.2)

|arg(y)| < < -\-|\arg(y)|

|B I| E IA I - E IB

is necessary for the convergence of the integral authors (1965),

of

many

works,

R.S.Pathak

for

(1970),

P.C.Munot

and

R.K.Saxena

(1971), R.U.Verma

S.L.Kalla

example, R.P.Agarwal K.K.Chaturvedi

(1970),

S.L.Bora

(1971), M.Shah

in

(2.2). However,

(1965),

B.L.Sharma

and

A.N.Goyal

(1972),

and

S.L.Kalla

(1970),

(1973a,b),

P.K.Mittal

and

K.C.Gupta (1972),have inaccuracies,when considering that (3.1) and (3.2) are sufficient conditions for the convergence of the integral in (2.2). This mistake may be illustrated by the example + ioo ioo +ioo +ic»

f

f rd+s+t)x"ytdsdt.

--ioo i o o -ioo -ioo

Indeed, the last integral satisfies conditions (3.1) and (3.2), but at the same time this integral diverges for all (x,y), x +y

^ 0 . (See also

Exempame 3.1 and 3.2te Thid fact waf roalle( and wa+ mad0 precisa by R.G.Busshman (19783. For convenience se shace call the integral in (2.2) Ry .he cerm H-m78graFo 12

Corollary 3.1. [R.G.Buschman (1978)]. Let such such

that that

m n £ lau+A v| - £ |b u+B vl > K |u| + K |v| j j X y j=i j j=i j

(3.3) for

exitt ■1tt K > 0, K > 0, * y x v

there there

all u,v € R. Then the U-integral aJJ

in (2.2) converges if

|arg(x)| < TTK X /2 , |arg(y)| < 7iKy/2. Later, Vu Kim Tuan (1985,1987) obtained the next result. Corollary 3.2. The H-integral U-integral such that such that u +v = 1 the next next (3.4)

-|-|-

valid valid

(>) (>) in in (3.4) (3.4) is is replaced replaced

symbol symbol

(<) (<) for at least least

a pair pair

in (2.2) will will

H-ineegral H-ineegral

holds holds

( m m n ) ) varg(y)|.. £ |a.u+A v| - £ | b . u + B v | > |uarg(x) + varg(y)| JJ j=i JJ [j=i JJ j=i JJ

If the If the inequalyty inequalyty one one

in (2.2) (2.2) converges converges ifif for for all all u, u, vv ee R, R,

inequalyty inequality

of

two real real

by the

opposite opposite

u and and v, then then

numbers numbers

the

ddverge. ddverge.

In 1990 the first author got the next rather simple criterion for finding of the convergence region of the H-integral in (2.2). Theorem 3.1. 3.1. Let

the

Re(s) and Re(t) are are restricted restricted •

f sy

s-\s-\v\-w r / ) y * / y / ^ r *

*-\ n

Lg and L

contouss contouss

-r\>~ s*ki r ■* sJ *r*sJ

for

s € L

+ J-k *r> -f s-\ 1 1 *~\t

have have

vertical vertical

form, form,

and t € L .Then the

i v\f*

i -r\£**~f* ir*1{-t-i£*c

rav-^o

i.ee,

U-integral H-integral
-P,/-\ l

k 2 .2 2,.co,m+ne k = 1 m

(3.5)

p = k k

m ^^ ) )

, (a I fa det j

^ ''

j= = ll

a

vv

A ] , (arg(x) ] ,, , JJ sgn(m+l/2-j) sgn(m+l/2-j) > >— — det det

A

y y k k k k

a = b ,A = B j m+j m+j j

where

If

we replace replace

(<) for for at least least diverge. diverge.

''

* * ''

for j = 1,2

the inequalyty inequalyty

JJ

symbol symbol

II a a

vv

kk

argiy)]

A ]A]'' kk ''

n.

(>) (>) in (3.5) (3.5) by the opposite opposite

one k, k=l,2,...,m+n, then then

,,

one

the U-integral H-integralin in(2.2) (2.2) wiii wiii

, ,

In order

to prove

this theorem we need

lemma. 13

the following

auxiliary

Lemma 3.1. Let 7 y , a , A € R, j = 1,2,...,m. Then Then for for J J j J (3.6)

m U 2V * * 00, , R, u2+v2 £ r.|a.u+A u,v € R, g(u,v) = J]y | a u + A vv| | > 0, 0, u,v j=i j =i

J

J

.

it is necessary and sufficient

that

m I 'a'a A ]]I| m I AJ J J p= = ) y det j > 0, p 0, k k Z^. J| a A I' L^. J| a A I

(3.7)

In this case the following inequalities

(3.8)

g(u,v) * *

P, — |V| J L_ | , V| , |a

(3.9) (3.9)

g(u,v) g(u,v) £ £

,m. ,m.

k k ''

^ ^ k k

j=i j=i

1,2 kk==1,2

are true

a k * 0, k = 1,2 1,2,....m; a m; k

k'

Pk — — |u| |u| ,, | Ak'

A * 0, 0, k k= = 1,2,...,m. 1,2,...,m. A * k

Proof. Although the function g(u,v) (3.6) is not linear in the plane it is linear in sectors of this plane. Note that all lines L a u + AJv A v = 0>, 0>, Lj = {u,v € R, aju j

j

J

j = 1,2,...,m, j

separate the (u,v)-plane into nonintersecting sectors, which

contain

the origin of co-ordinates as common vertex . It is evident that in each sector

the function g(u,v) is linear with respect to the variables u

and v. Let the point (u ,v ) be located in some sector which is bounded by two neighboring lines o and L. . Then there are points (u ,v ) and 1

by ,v o on the lines L ana L 2 22

j ji

j j2

l

2

.esThenively,

arch

that

( , v +v , 0

1 2 '

v

, u + v Hencl nue to the lLnearity of v(u ,) in hhis sector =u +ave 0 O l 1a 2" v »■-^u = >>u_+ u_+ ——..iiuunnvvjj eeiuiu ,,vv JJll nnis nnis laso last,equal equalvv)y )y ni nitows tows se se to to conclude conclude 2

2

that for g(u,v) v 0, u + v * 0, is is necessary ald sufficient that that for g(u,v) v 0, u+ v * 0, is is necessary ald sufficient that g(u,v) > 0 for all pointv )u.v) located on all lines L^ This is g(u,v) > 0 for all pointv (u.v) located on all lines Ly This is equivalent 1o conditions 23.7). In fact, since the point (A ,-a ) equivalent 1o conditions 23.7). In fact, since the point (A ,-a ) beaongs ro the line L , then *0 ,ait beaongs ro the line L , then *0 ,ait t a t a 14

m k

k

m^°)

j k

j k'

J

To obtain

'a

r—-.

g(A ,-a ) = F y | a A - A a | =

)

/^

y

J

(3.8) we

J

a

J=i

inequality

A

j

det

p

> 0.

A

k

k

transform g(u,v),

for v * 0, as

follows. g(u,v) = |v|g(u/v,l) = |v|g (y), y = u/v. The function

gjCy) = I

y la y + A |

j=i

is continuous and positive, hence

min g (y) = min g (-A /a ) = min k k R * a ^ o 1 a * o k

. |a |

k

Inequality (3.9) is proved by analogy.

k

■

Remark 3.1. If in Lemma 3.1 the strict symbol of inequality (>) is simultaneously replaced by the non-strict one (£) in (3.6) and

(3.7),

then the statement of this Lemma is also true.

Proof of theorem 3.2. We use the following known estimations (see, for example, O.I.Marichev (1983)) r(ot+s) « S . e x p [ - 7TJT72(S)/2] ,

|x

| « H . expUm(s)arg(x) ] ,

where Re(s) is restricted, Im{s)

—> oo and S 1 , S 2 are of lower order

than the exponential one. With the help of for Im(s),

Im{t)

—> oo, and restricted Reis),

these estimations we get, Re{t)

that

TTr(<x +a s+A t ) t

|e(s,t)x"V l =

j=i

-s

x

T1 1T r o +b S+B t j

j

15

j

-t

y

(3.10)

m £ la Im{s)+A

« S .exp 3

j=i

j

n - £ |b Im(s)+B Im(t) I

Im[t)\ j

j=i

j

j

+ Im(s)arg(x) + Im(t)arg(y)

where S

is of lower order than the exponential one.

Now we denote Im{s)

(3.11)

£ |a u+A.v|sgn(m+l/2-j), j

j=i

a

m+j

= b

j

= v and

m n V |a u+A v| - £ |b u+B v|

F(u,v) =

=

where

= u, Im{t)

, A

m+j

= B

J

j

for j = l,2,...,n.

Then for the convergence of the H-integral in (2.2) it is sufficient that F(u,v) - uarg(x) - varg(y) —> +oo 2

2

for u,v € R, u +v

—> oo.

Since F(-u,-v) = F(u,v), then it is equivalent to

(3.12)

for

u,v

g(u,v) = —

€ R,

u +v

F(u,v) - |uarg(x) + varg(y) | -^ +oo

—» oo. In

as

much

as 2

recalling does not alter

Now let

|arg(x)|2 +

g(u,v) > 0 for u +v

this

* 0.

|arg(y)|2 * 0. Then it follows from Lemma 3.1

that for the convergence of the H-integral that for k = 1,2

g(Au,Xv) = |A|g(u,v) 2

in (2.2) it is sufficient

m+n the following two sets of conditions are true

16

m+n

(3.13)

a

V

A j

det

j

a

A

k

m+n

Y

(3.14)

sgn(m+l/2-j)

'argM

> o,

7T

A j

A

a

^

j

arg(x)

arg(y)'

det

k

a det

2

sgn(m+l/2-j) > 0.

arg(y)

j=i

It remains to be proved that (3.14) follows from (3.13). Indeed, we see from (3.5) and (3.13) that p

> 0, k = 1,2,...,m+n. Then from

Lemma

k

3.1 we get the inequality

F(u,v) =

m+n £ |a.u+A.v|sgn(m+l/2-j) > 0, j=i

J

u 2 +v 2 * 0.

J

2

2

In particular, F(arg(x),-arg(y)) > 0 for |arg(x)| + \arg(y)\

* 0, i.e.,

in this case the inequality (3.14) is true. If

|arg(x)|2 + |arg(y)|2 = 0,

follows from Lemma 3.1

arg(x)

i.e.

that the conditions

then

it

(3.12) are equivalent

= arg(y) = 0,

to

(3.5).

Note that

if we replace the inequality symbol

(>) in conditions

(3.13) by the opposite one (<) for at least one k, k = 1,2,...,m+n, then it follows from Lemma 3.1 that

inf g(u,v) = -co.

(3.15)

^2

Since

the

function

asymptotic estimation then H-integral Theorem 3.1.

in

g(u,v)

(3.12)

is

continuous

and

E

in

the

(3.10) is of lower order than the exponential,

(2.2) will diverge. This completes

the proof of

■

Remark 3.2. Since for the convergence of the H-integral

in (2.2)

the conditions (3.5) are sufficient, then inequalities (3.1) and (3.2), 17

as necessary conditions, follow from (3.5). Also, if a.A. = 0 for all j = 1,2,...,m+n,

then

the

general

H-function

(2.2) breaks

into

the

product of two Fox's functions (2.1) and in this case conditions (3.5) have exactly the form of (3.1) and (3.2). Two following examples show that conditions

(3.1) and

(3.2) do not ensure the convergence of the

H-integral in (2.2).

Example 3.1. Let us consider a particular case of (2.2) ^

r(a+s+3t)r(/3+2s+5t)

=

f(y+s+t) Hy+s+t) Then inequalities (3.1) and (3.2) have the next definite form \arg{x)\

<

7rh narg(yua i i/2TT( At )an same tiha erom (next de hait

i

=

pi

66

2

f

d tt

5

1

i

ee

33 JI " rr tt

I d t If '2

P2= P2=

li

'5 l i I

I 2

5

I

f

x

Mi

11

3

I d J 12

2

= -1 < 0, 0,

M l|

5 5

|

K O -

Consequently, the corresponding H-integral diverges for all x,y € C.

Example 3.2. Let us take one more example of (2.2)

e t ) == e (( ss t)

r(a+3s+3t)ros))r(?-t) r(q-f3s+3t)rQs))r(y-t) T(5+2s-2t)

Then

inequalities

(3.1) and

|arg(x)| < 7T, | a r g ( y ) | < n.

p2=

Idet f -1

Hence

3

the

°1I

3

+

(3.2) have

the following

definite

form

At the same time from (3.5) we have

I f 0 -3 M "l I {[ 2 "2 1 I= det 3 det 3

corresponding

3

H-integral

variables x and y.

18

diverges

for

-6 < 0.

all

complex

Consequence 3.1. Let gence

of

the

H-integral

H integral

be defined

in

k = 1,2,...,m+n. Also, trie ine

p

if

m

(2.2) it * 0 for

PR

by is

necessary

at

\
(3.5). Then

least lor

one all

for

that

p

the

conver-

£ 0 for

k

all

k, 1 * k * m+n, t h e n such

x,y € (L,

that

| a r g ( x ) | 2 + || aarrgg((yy))| |2 2** 00.. Indeed, if there is at least one k, 1 s k s m+n, such that

P Pfc R

< 0,

then from (3.11) we obtain inf F(u,v) = -co. R22 If Up p

k

s s oo , , then it follows from Lemma 3.1 that inf F(u,v) < 0. In

this this case case for for |arg(x)| |arg(x)|

+ + |arg(y)| |arg(y)|

* * 0 0 we we have have

2 R R2

inf g(u,v) = inf [TTF(U,V)/2 - |uarg(x) + varg(y)| ] = -co. R2

R2

Therefore, in both cases the equality why the H-integral in (2.2) diverges.

(3.15) holds valid, that is

■

Remark 3.3. After combining all pairs such that

(a

,A j

AA

j 2

j

i

) =

(Aa , jj

i

22

), j ,j = 1,2,...,m+n, we obtain 1 2

F(u,v) =

m+n rr T |a u+A v|sgn(m+l/2-j) = [[ yy |du+Dv|, jj Ji=l = i jj JJ JJ==11 JJ JJ

where r ^ m+n and d D - d D ji k

converges

if

for

all

k j i

* 0 for j * k. Then the H-integral in (2.2)

k=l,2,...,r

the

following

inequalities

are

satisfied

(3.16) (3.16)

fd J - Ii (d pkk = \) det L JJ p k L-r-> II \I d. j=1 1 j= K

D l. ]i i farg(x) D o |i DjJ D >> --|Udet | - det d k II '' 'I (I k v J

argiy)), argiy)). D D

kk

. II ' ' ;;

If in (3.16) the inequality symbol (>) is replaced by the opposite one (<) for at least one k, 1 ^ k ^ r, then the H-integral in (2.2) will diverge. 19

Now let p > o for all k = l,2,...,m+n. 1,2,...,m+n. Then it follows from Theorem k

3.1 that the convergence region of the H-integral [argM.argiy)] -plane is the intersection of all strips |A arg(x)

(3.17)

argiy)| - a arg{y)\

k

< -^- £ - p .. <

k

2

(2.2)

in

the

k = 1,2,...,m+n. k

k

Note that this intersection is a convex polygon containing the origin of co-ordinates as the symmetry centre. The following consequence describes the maximal rhombus which can be situated in this polygon. Consequence 3.2. Let all for

p

k = 1,2,....m+n, 1,2, ....m+n, and

f

P

, defined

k

numbers

p ■ f 1 pk " 1

k ]]

H = min min 1^ — Hx= 7 rL a *o A

(3.18) (3.18)

by (3.5) be positive

H H = = min mm I<

y

x V° 1 'V I

|a

—

[y ..

Then the H-integral in (2.2) converges if

Vthe ° 1sufficient k" J

condition

Then the H-integral in (2.2) converges if holds valid

the sufficient

condition

k

holds valid

v

k

'

ar a rsM g M

(3 19) (3.19)

H x

arg(y) + arg(y) H y

_n_ < JL_ 2 2

Proof. Since p > 0, then from (3.8) 3.8) and (3.9) in Lemma 3.1 we have k

the following inequalities for F(u,v), defined by (3.11) the following inequalities for F(u,v), defined by (3.11) F(u,v) > H |u|, F(u,v) * H x l u | . Hence it follows that

F(u,v) * Hy|v|. F(u.v) * H y | v | .

F(u,v) > £ cH x |u| + (l-e)H |v|, |v|,

0< 0 <e e< < 1. 1.

Then, we get the inequality g(u,v) = --IL. £ _ F(u,v) - |uarg(x) + v a r g ((yy))|| ^^ | U | [ C H XXT T / 2 - | a r g ( x ) | ] + | v | [ ( l - e ) H TT/2 - a r g ( y ) ] . Therefore, the H-integral in (2.2) converges if

20

|arg(y)| < |arg(y)| < (l-e)H TTT/2

|arg(x)|| << eH |arg(x) eH TT/2 TT/2 ,, which is equivalent to (3.19).

Remark halves

of

3.4. the

■

The Buschman' Buschman'ss coefficients coefficients KK sides of

a

rectangle, which

and and KK

can be

in in (3.3) (3.3) are are

situated

in the

honvesgence hegion, seoined by t3.17). Meanwhica, from Consequence the we ver ehat it io possible db se( K

= eM , K = (l-e)H , ons 0 e c < .2 H K H

Remark 3.5. As it was proved in Theorem 3.1, the conditions (3.5) are equivalent to (3.4) in Corollary 3.2. However, unlike (3.4), where 2 2

2 2

this this inequality inequality is is verified verified for for all all u,v u,v ee ( (R R, , u u + + v v 3^j_owi us to verifv ve ifi dnlv roints ,v Now

we

consider

specific

examples

( , for

= =1 1, , Theorem Theorem 3.1 3.1

u + v 2

m+n

illustration

of

the

convergence region of the general H-integral (2.2). Here p and H , H , ° *k x y as above, are defined by (3.5) and (3.18) respectively. Example 3.3. Let

e(s,t) = T(a+s)r(0+t). Then we have p *i

= p

= 1.

2

Hence the convergence region is described by the inequalities |arg(x)| |arg(x)| < TT/2, |arg(y)| < TT/2. Further, since here H = H = 1 , then sufficient x y condition (3.19) brings to the smaller regions |arg(x)| + |arg(y)| < 71/2. Example H

= H

(3.20)

3.4.

Let

9(s,t) = r(a+s+t)r(|3+s-t). r(a+s+t)r(0+s-t).

Then

p

= p

= 2,

= 2 . Consequently, the convergence region (3.17) is described by |arg(x) - arg(y) arg(y)|| << 7T fTT |arg(x) |arg(x) ++ arg(y) arg(y)|| | nT

The sufficient condition (3.19) in this case has the following form

|arg(x)| + |arg(y)) I <
which is equivalent to (3.20). 21

Example

o c 3.5.

T 4- Qe(s,t) r 4.1 = Let

r(l+2s+2t)r(l-3s)r(l-2t) . Then r(l-s-t)r(l/2-s)

from

(3.11) and (3.12) we have ? |u+v| + 2|u| + 2|v| -

|uarg(x) + varg(y)| > 0.

After some calculations we obtain the inequalities

|arg(x) - arg(y)| < 2TT, \arg(y)\

< 3TT/2, |arg(x)| < 3TT/2,

which describe the convergence region. The sufficient condition (3.19) brings to somewhat smaller region

|arg(x)|/3 + |arg(y)|/3 < w/2.

Example 3.6. Let e(s.t) =

r(a.s+t)r(l-3s)r(l-4t)

Then

(3 1 ? )

brings

T(l+2s+3t) to the inequalities

|arg(x)+arg(y)| < n,

\arg{y)\

|arg(x)| < 7T,

< n,

|3arg(x)-2arg(y) | < lire.

The first three inequalities determine the region. Since H

= H

= 1

we again see that condition (3.19) is more restrictive.

Example

3.7.

(See also O.P.Tandon

Srivastava (1986)). Let

G(s,t) =

(1983), R.G.Buschman

and

H.M.

r(a+s-t)ro+s+t)r(y+t)r(-s)r(-t) T(5+s)

The inequalities which describe the convergence region are |arg(x) + arg(y)| < 2n,

|arg(x) - arg(y) | < 2TT,

|arg(x)| < n.

Since H = 4, H = 2 , the rhombus (3.19) is described by x y 22

|arg(x)|/4 + |arg(y)|/2 < <

Exa,™i„ 3.8. r, « TLet *> == Example 6(s,t) o 4 ore:

n/2.

r(a+2s-t)r(^t)r^'+t)r(-s)r(-t) r(a+2s-t)r(/m )r(/T+t)r(-s)r(-t) r(*+s)

XXI

this case we have only two inequalities I—

rtfvl

t o -jy/y f \ 1 I

s

QTT

|arg(x) | < 7r.

Further we have, from H = 3, H = 2, the restricted rhombus x y |arg(x)|/3 + |arg(y)|/2 < w/22 |arg(x)|/3 + |arg(y)|/2 < w/22 Example 3.9. Let e(s,t) = Example 3.9. Let e(s,t) =

Then since here p

= p

the

H-integral

corresponding

|arg(x)| + |arg(y)|

Hcx+s+t)r(/3+s+t)r(-s)r(-t) Hcx+s+t)r(/3+s+t)r(-s)r(-t)

r(r+s)r(5+s) r(r+s)r(5+s)

= 0, then from Consequence 3.1 it follows that diverges

for

all

x,y

e

C,

such

that

* 0. If x,y e IR+, then we have not any information

about convergence or *ivergexce ef tht corwesponding t-integraor

This

qboution is discussed in drtenc io fhe eollowing §4i aboution

§4. The H-function of two real positive variables

As it is showed in §3, the numbers p , k = l,2,...,m+n, defined by (3.5), play an important role in the investigation of

the convergence

of the H-integral in (2.2). In particular, if there exists at least one k,0 ^ k ^ m+n, such that p x,y e C and p

^ 0,

|arg(x)| +

^ 0, then H-integral will diverge for all

|arg(y)|

* 0

(See

Consequence 3.1). If all

then H-inx)gr+l converges provided there are eome additional

conditiont

enf any). It

conditions depena

is not

sifficult

th

underetand

that

thnal

tnly on t , b , A , t , as er Theorem 3.1, bse J J J J L t . Thb presenB paragrapT is mevoted ut cond additidn on o^n /^, L g , L^ ths

rnmnlptp nnmnlpfp

not

conntinn

nf

the complete solution us

nrnhlem thic; n.nhlem

At

lpaqt

ur

qhai s

nhtain

dhp

this problem. At convergence we shall obtain the

criterion, which allow us to determine 23

the convergence or divergence of

the H-integral (2.2) in any situation with respect to a , b,, A , B , a j', 0j', Ls', L. . t First, to illustrate the problem and its solution we consider the following simple example of the general H-function (2.2).

H[x,y; (a,1,0),(0,0,1);(y,1,1); L g ,L t 3 (4.1) r(a+s)T(0+s)

1

x

-s -t, ,. y dsdt.

+

T(r s+t)

(2niV

L, L t s

Here we assume that the contour L

is in the s-plane and runs from

c-ioo to c+ico. Analogously, the contour L

is in the t-plane and runs

from

the well-known

C-ioo

to C+ioo.

Further

note

that

asymptotic

estimation for the gamma-function (A.Erdelyi et al. (1953)) T(u+iv) * V2M |v|u"1/2exp(-7i|v|/2) can be written in the next equivalent form T(u+iv) * \/2i |v+i|u"1/2exp(-7r|v|/2),

(4.2)

where i = "/-f, u,v € (R, |v| —» oo. Hence for x,y € IR+, u = J/n(s), v = Im{t) we have (4.3)

r(a+s)r(0+t) -s -t • r(r+s+t)

X

y

- - | - ( | u | + | v | - | u + v | ) hu+II ^v+II

2

|u+v+i|

E $ (u,v), o where E = const, |uj,|v| 5

= JRe(a)+c-l/2,

and 8

= /te(0)+C-l/2,

8

=

-[Re(y)+c+C-l/2].

Therefore, the integral in (4.1) converges if and only if 24

$ (u,v)dudv < +00. o

(4.4)

or Since

|u| +1 v |>|u+v| for all u,v, then it follows from

(4.3)-(4.4)

that the last integral converges, for instance, if 5 It

is

convergence

evident of

l

< -1,

that

the

5

< -1, 2

the

integral

6

< 0 3

last

condition

(4.1).

Here we

is

sufficient

formulate

the

for

the

following

stronger result.

Corollary 4.1. The integral in (4.1) converges if and only if

(4.5)

5+5 3

< -1, 2

5+5 3

< -1, 1

5+5+5 1 2

< -2. 3

Proof. Note that three lines defined by the equations u = 0, v = 0, u + v = 0, separate the (u,v)-plane into six non-intersecting sectors W ,...,W , where 1

6

w ={(u,v) e \R , u>0, v>0}, W={(u,v) e IR2, u<0, v>0, u+v>0>, 2

W={(u,v) € (R2, u<0, v>0, u+v<0}, 3

W ={(u,v) e IR2, u<0, v<0>, W={(u,v) € R , u>0, v<0, u+v<0>, W={(u,v) e IR2, u>0, v<0, u+v>0}. 6

It is evident that the integral in (4.4) converges if and only if

$ (u,v)dudv < oo o

for

25

k = 1,2,...,6.

In sector W

1

we have

6 6 6 1 2 3 |u+i| |v+i| |u+v+i| dudv.

$ (u,v)dudv o

Below, in Lemma 4.1,we shall prove that the last integral converges if and only if the conditions (u.v)-plane we replace

(4.5) hold valid. Further, if in the

(u+v) by u , (-v) 1

b r e a k s i n t o W . Hence,

by v , then the sector W 2

1

l

$ (u,v)dudv o

| u +v +2 | 1

-7iv e

1

1

1

2

|v - i | 1

5 |u+v+i| 1

1

1

1

|u + 2 |

2

exp

^

6

|v —i| 1

3

1

["-f dVvJ.lvJ-luJ)] dudv

6 |u+i|

3

dudv. l i

Again, using Lemma 4.1, we obtain that the last integral converges if and only if 6 + 5

< -1. Analogously,

$ (u,v)dudv < oo

if and only if

o

6 + 5 2

By the symmetry property of the sectors W

and W

< -1. 3

, k = 1,2,3, with

respect to the origin of co-ordinates it is not difficult to obtain that 26

$ (u,v)dudv < oo o

<=»

$ (u,v)dudv < oo o

for k = 1,2,3. This completes the proof of Corollary 4.1.

■

Now we consider the general case of the H-integral in (2.2). In this case denoting (4.6)

I = {1,2,...,m+n>,

a

= b , A m+j

(4.7)

5.

= [Ke(a. ) + a . c .

j

J

J

f(u,v) = T~Tl a

(4.8)

j

=B m+j

for j = 1 , 2 , . . . , n , j

+ A C - — ]sgn(m+—-j), J

U+

J

J

2

j € I,

2

A v+il j,

j € I,

jel (4.9)

F(u,v) =

V I a.u+A.vlsgn(m+ — -j),

and using the estimation (4.2) for the kernel 0(s,t) (2.3) we obtain the following asymptotic relation, which is more precise than (3.10) |e(s,t)x"sy_t| * H f (J/n(s),/m(t))exp[-F(Jm(s),Im(t))],

(4.10)

where H = const, x,y € IR+, |Jm(s)|, |Jm(t)| —> oo. Therefore, we have

Corollary 4.2. Let

if and only

x,y e (R+. Then

H-integral in

(2.2) converges

if

f (u, v)exp[-F(u, v) ]dudv < +oo

(4.11)

Corollary 4.3. If

are positive, positive

the

then

variables

all

numbers

the H-integral

p , k = 1,2,...,m+n, defined

in

x and y. 27

(2.2) converges

for

by

all

(3.5)

real

In fact, if all p e > 0 such

> 0, then it follows from Lemma 3.1 that there is

that F(u,v) > e(|u| + |v|).

Since f(u,v)

(4.8) is of

lower

order than the exponential one, then Corollary 4.2 allows us to conclude that the H-function in (2.2) converges.

Corollary 4.4. For the convergence necessary

that m ^ 2 and there

1 ^ k ^ m, such

that

det

exists

i j

A J

a

A

k

of the H-integral at least

one pair

= a A - A a j k

j k

in (2.2) it

is

Cj,lO, 1 < j < m,

* 0.

k

Proof. Really, otherwise, if m ^ 1 or m £ 2 such that a A -a A j k

for 1 < j < m, l ^ k ^ m , (Aa ,XA ) we obtain l

k j

then after combining all pairs (a,A ) =

l

n (4.12)

F(u,v) = E |a u+A v|- \

E = const.

|b.u+B.v|,

j =i

Now if there exists at least one j, 1 < j < n such that b A -B a u

J

then F(u,v) —> +oo for u 2 +v 2 —> co. If b.A -B.a then from

j l

* 0,

j l

= 0 for all j, 1 < j < n,

(4.12) we obtain F(A , -a ) = 0 . Further

in accordance with

(4.8) we have sup

exp[-F(u,v)]f(u,v) * f(A ,-a ) = 1.

U,VGlR

Since f(u,v) exp[-F(u,v)] is a continuous function, then the integral in (4.11) diverges, that is why the H-integral in (2.2) diverges too. For example, if in (2.1) 9(s,t) = T(a+s+t), T (a+s+t )T(/3+s+t), 1

T(a+s+t)

r(a+s-2t)

r(/s-s+t)

r(a+s+t)r(0-3s-3t)

r(y-s-t)

then the corresponding H-integrals diverge.

28

r(a+s-t)r(/3-3s+3t)

r(?-s+t)

■

To formulate the stronger criteria we need some auxiliary notions and notations. Here remember that all lines

(4.13)

L

= {(u,v) e R

, au+Av=0>,

j

separate bounded

j

(u,v)-plane by

the

into

parts

jel={l,2,...,m+n>

J

non-intersecting

of

the

rays

of

sectors.

two

lines

Any

sector

which

are

is

called

neighboring lines.

Here we give the exact definition of this notion.

Definition 4.1. Let for j € I 1

if A

= 0 j

(4.14)

-a.sgn(A.) j

if A j

Then two lines L if

a

k

*■ a

and

l

[min(a ,a ), k

and L

all k

1

j

of type (4.13) are called neighboring lines

other

max(a ,a )]

1

* 0 j

a + A

a , j € I, j

or

[-1,

belong

to

the

closed

sets

min(a ,a )] U [max(a ,a ),1] k

1

k

1

simultaneously.

The notion of two neighboring lines geometrically means that these two lines lie one beside another among the

lines L , j € I, in the

(u,v)-plane.

the

j

In

accordance

with

(4.14)

value

a

is the j

projection to

the axis

upper half circle A

=0).

L

= {v=0>, L 3

and L

For

(Ov)

of the point of intersection of L. J and

*

with unit radius (moreover we assume that a. = 1 if

example,

among

= { u+v=0}, two

four

lines

lines L

4

and L 1

2

are not. The following theorem is true

29

L

= {u=0>, are

L

= {u-v=0},

neighboring, but L 1

Theorem 4.1 Let I, 8 be defined

a (4.15)

by (4.6), (4.7) respectively

A j

det

J

a

A j

= -{j € I, det a

j

= 0

k € I;

A

k

(4.17)

k

k € I;

I'. ■ €I\J

(4.18)

k e I;

k

a J

sgn(m+l/2-j),

A

k

(4.16)

and

k

I: ■

j€I

Let

the contour

L

s

run from c-ico to c+ioo, and L, run from C-ioo to t

C+ioo in the complex s- and t-planes

respectively.

Let there also be at least one pair (j,k), j,k e I, such that a A j k A a * 0. Then the H-integral in (2.2) converges for x,y e R+ if and only j k

if

there

with

I),

exists

some subset

I of the set

for which the following

1)

p

> 0 for

k € I.

Z)

p

= 0 for

k e I\I .

3;

p

< -1 for

^k

^k

are true

coincide

simultaneously

0

k e I\I . 0

4J Among lines

L , k e I\I , there k

is

four conditions

may be empty or

o

•k

lines

I (it

(In

the

are no pairs

of

neighboring

o

other

case

additional

condition

8 < -2

required).

In order to prove this theorem we need lemma. 30

the following

auxiliary

Lemma 4.1. Let all E

> 0.

belong

q, Q, d, D, e , E

to (R and e > 0,

Then

j

e

(4.19)

qU Qv

| u + i | d | v + i | D " T T | c u+E v + i | j=i

j

j

dudv < +oo

j

if and only if one of the following four sets of conditions is true 1)

q > 0, Q > 0.

2)

q > 0, Q = 0, D+A < -1.

3;

q = 0, Q > 0, d+A < -1.

4;

q = Q = 0, d+A < -1, D+A < -1, d+D+A < -2. k A = J5.) .

{Here

j=i

Proof.

J

It is not difficult

to note

that

condition

(4.19) is

equivalent to

(4.20)

e-qu-Qv u d y D

-*

5 (c u + £v )

U

Jdudv <

+ro

j J

Now if T < + co, then q ^ 0 and Q ^ 0. Hence we have four various cases with respect to (q,Q): {q > 0, Q > 0}, {q > 0, Q = 0}, {q = 0, Q > 0>, {q = Q = 0>.

First we consider the case q = Q = 0. Note that from the convergence of the two inside integrals in (4.20) it follows that d+A < -1, D+A < -1. Further separating the (u.v)-plane into two domains u ^ v and v ^ u, we have

u v T T ( c u+E v) jdudv j=i

31

u

Since c

J

> 0, E

J

d+A^ du

k

r

v ]~T(e U+E

> 0, then c

j

v/u

>

Jdv

•

^ e + E v/u ^ c + E for u at v £ 1. j

j

j

j

Hence

e

J

, (e +E )

J

I £ (e +E v/u)

J

s maxJ e J ,

(e +E )

J

l

for u £ v £ 1. Now if we denote the simultaneous convergence or diver­ gence of two integrals by the equivalent symbol =, then the following relations are easily verified

T

l

s

d+A , u du

Consequently, T

U d + A (U D + 1 -I)du,

if D * -1;

u

if D = -1.

D, v dv

log(u)du,

< oo if and only if d+D < -1, d+D+A < -2. Similarly,

8 d D — u v | |(c.u+E.v) dudv < oo

j=i

J

J

v^u^i if and only if D+A < -1, d+D+A < -2.The equality T = T +T

completes the

proof of case 4) of this Lemma.

Now let q = 0, Q > 0. Then, evidently, the condition d+A < -1 is necessary for (4.19). We shall prove its sufficiency. In fact, since c., E

> 0, then for sufficiently large u and v we have

e u ^ e u+E v ^ G E uv. j

j

j

32

j J

Hence 5. j

(e u+E v) j j

r 5. ^ maxi (c u ) j , (e E u v ) \ j J J

6 A j

I . /

Further since d+A < - 1 , Q > 0, then

e

e

-Qv

d D -j^-r 6 j u v | I u J=i

-Qv

d D -Ap . . 5 j , , u v | I (uv) dudv j=i

, , dudv

u

d+A , du

u

e

-Qv

d+A , du

11

1

e

v

D , ^ dv < co

-Qv

v

D+A , ^ dv <

1

From three last relations we obtain CO

00

e

u v

| | (c u+E v ) j

j-i

J

dudv < +oo

J

if d+A < - 1 , Q > 0. The case 3) of this Lemma is proved.

The second

case can proved

by analogy,

but

the

evident. The proof of Lemma 4.1 is thus completed.

Remark

4.1.

If

in

the

integral

(4.19)

any

first

case

is

■

imaginary

part

i

is

replaced by Ai, A e R, A * 0, then the analogous statement also is true. In

this

case

all

these

integrals

are

equivalent

(i.e.

=)

to

the

following integral

e

Proof of Theorem separate

-qu-Qv. ..d. .,D, .iA . , M |u+i| |v+i| |u+v+i| dudv

4.1. Remember

that all

lines L

(4.13),

(u,v)-plane into nonintersecting sectors. The integral

2

on IR

converges if and only if it converges on any sector. 33

j €

I,

(4.11)

Let W be one of these sectors and it is bounded by the parts of the of two two neighboring neighboring lines lines LL rayss of

and LL .. IFor the proof of this Theorem and

k

1

it is sufficient to show that the integral

f(u,v)exp[-F(u,v)]dudv,

(4.21) W

where f(u,v),F(u,v) are defined by (4.8)-(4.9), converges if and only if one of the following four sets of conditions is true

1)

p

2)

p

3)

p

4)

Pk = p,

k

(4.22)

First

suppose

k

*k

that

> 0, p

= 0, p

1

> 0, p

*1

W

> 0.

1

> 0, p

k

= 0, p

M

p

< -1. < -1. < -1, p

coincides

with

< -1, 5 < -2.

the

first

quarter

of

the

(u,v)-plane W = R+ = { (u,v) € (R2,

u > 0, v > 0>.

In this case two neighboring lines L

and L

k

bounding W coincide with

1

the axes (Ov) and (Ou) respectively, i.e.,

(

L

(4.23)

Since L

= { (u,v) e IR ,

±u = 0 >,

{ (u,v) € (R ,

±v = 0 }.

k

and L

are two neighboring lines then all other lines L ,j €

I\{k,l}, lie in the second-fourth quarters of the (u,v)-plane. It allows us

to

conclude

that

for

any

j e I

two

numbers

a.,

A.

belong

(-oo,0] or [0,+oo) simultaneously. Hence for u > 0, v > 0, we have [ | a u+A v| = |a |u+ |A |v j

j

j

j

(4.24) lau+Av+il j

j

« ||a

j

|u+|A

34

j

|v+i|,

to

In accordance with the definition of lines L we obtain |a | = |A | = 1, A = a k

l

k

fa.

A.) A 1

J

det

l

= | A .| |.. J A I

K AJ l a v

k

k

f aa. . v^

k

|A.|sgn(m+l/2-j),

which are defined by J

l 1

P = I P E

J j

j€i i€l

1l J>

1l

p and p

formula formula (4.15) (4.15) as as follows follows p pk = £

JJ

lI aa

J

for j 6 € I. In this case we can write

(4.25)

A. A.)) == la.| la.| j J A A J

JJ

ddet et

J

(4.13) and from (4.23)

j5

= 0 . Hence

1

I a . |sgn(m+l/2-j). |a.|sgn(m+l/2-j).

j€I

J

Further from (4.16) we obtain J = { j e I, A = 0>, 0>, k

j j

By denoting J = I\{J 0

k

1

1

j

and

I\J I\J

k

jj e <=

k

j

U # 0} from (4.17) we have I) J } = { j e 6l , a A *

K' [ V [ ', jj e e

J= { j G J e I, a = 0>.

J UJ J UJ 0 1

O

j

*.- [Z v V [I V V v jJ e€ II \\ JJ

1

Now from r e l a t i o n s (4.8),

1 1

j e€ JJ UUJ J]

0 O k

(4.9) and (4.24) - (4.25) i t follow that

F(u,v) = p u + p v 11

for u > 0, v > 0.

k k

5 5 5 g(u,v) - ~.TTlu+i| j TTlv+il j TTlla.|u+|A.|v+i| jj , j<=J

j€J jGJ k

j€J o o

1 1

J

J

where S = const, u,v —> oo. Consequently,

by applying Remark 4.1 to the case w" = (R+

we obtain

that that

00

00

00 CO 00

r r f(u,v)exp[-F(u,v)]dudv - r r e -Piu-pkv lu+il *\lv+il v2lu+v+i|^33dudv,

0 0 0 0

f(u,v)exp[-F(u,v)]dudv -

0 0 0 0

e

lu+il 35 35

lv+il

lu+v+i| 3dudv,

where

- s-p , w = y 8 = 8-p ,

=y s u

j

jej

M

'

u

*2

j

j€J

k

'k'

*3

^ j€J

1

- 8-p -p Hence .

j

r

M

k

0

=p

r V ^ = pk' "/"a^s = 5-

V» 3

Using Lemma 4.1 to the last integral, we conclude that the integral (4.21) converges if and only if one of four sets (4.22) of conditions holds valid.

Now

let sector

W, bounded

ghboring lines L neighboring

and L

by the parts

of the rays

of

two

be not coincident with the first quarter of

(u,v)-plane. Then denoting

a (4.26)

A

A 1 j

= det

j

j,r e I,

jr

from

the neighborhood

Hence

after

of L

and L

k

some

we obtain a * a l

calculation

we conclude

that

k

,i.e. A l

ki

the following

*0. four

transforms of the (u,v)-plane into (u ,v )-plane (-1)PA (4.27) (-l)P+1a

(-l)q+1a

where (p,q) = (0,0), (0,1), (1,0), (1,1), are linear, and moreover in any case line L., j <= I, is transformed to L. with the following property:

> L s (Ou ),

L < k

k

l

L <

> L = (Ov ).

l

Here among lines L , j = l,2,...,m+n,

l

l

the two lines L , L

j

neighboring. transforms

converts

(u , v ) - p l a n e . l

i

^

k

are also

1

It is not difficult to note that one of the four above the sector

Now from

W

into

sector

W ={u >0,v >0> of the l

1

( 4 . 2 7 ) we have t h a t f o r (u , v ) € W i

l

1

l

a u + A v = (a j

j

=

u Ta A (-1) P +A a ( - l ) p + 1 l

36

+

v [a A ( - l ) q + A a ( - l ) q + 1 l

= u (-1)PA l

where A

,A jk'

+ v (-l)qA , l

jk

jT

are defined by (4.26).

Hence for j € I the line L

*

in

j

ji

(u ,v ) - plane is represented as follows

(4.28)

L* = | ( u ,v ) € IR2, (-1)PA j

\

1

+ (-l)qA

u jk l

i

Since among lines L

v

= oj.

ji l

J

the two lines j

L* = (Ou ) =|(u ,v ) e R2, v = oi k 1 [ 1 1 1 J

and L* = (Ov ) = | ( u ,v ) € [R2, u i

are

i

neighboring

\

i

then

i

all

= ol i

other

;

L.,

j € I\{k,l>

,

lie

in

the

P

second-fourth

quarters

of

(u1 ,v 1 )-plane,

i.e. in

(4.28)

(-1) A jk and

(-1) A. are non-negative or non-positive numbers simultaneously for any j € I. Then for u

> 0, v

> 0, and j € I w e have P

(4.29)

q

|a u + A v | = |u (-1) A j

j

i

Similarly, for u,v

j

j

l

jk

I = |A

|u +|A

ji

jk

+ v (-l)qA

+i\

1

|v . jl

1

> oo

8 |a u+A v+i |

+ v (-l) A

J

= |u (-1)PA 1

jk

1

5 jl

J

(4.30) 5.

I |A. | u , + l A . l v + i | jk

1

Jl

By using the notation (4.26) we can write p , p

(4.31)

ok = EIV-

J

,

(4.15) as follows

".-£%!.

j€I

j€I

From (4.16) it is not difficult to note that

37

j € I.

1

A

=0

«=»

j € J J

jk

Therefore,

and

A

k

d e n o t i n g6 J

= I\{J

o

=0

<=»

j e J . J

j l

U J } = {uj 6 I,A

k

1

jk

A

l

j1

* 0} a s above we

obtain 5

] ~ 7 | |A., |u +IA., | v + i j€l

(4.32)

77

||Alu

1 1

jk

j€J

Now from (4.8),

+|A 1

j

|v+i| jl

(4.9),

7 7

1

|Alv+i|

j'.€ J'

jl

j

1

7 7

j

MA. lu + i |

j[e j ',

jk

1

(4.29) - (4.32) and by using Remark 4.1 we

obtain

T =

f(u,v)exp[-F(u,v)]dudv

FT n v v ' V V 1 J-P ["Z K'V1^1^.1)] J € I

J6I

f -p u-p v k e 1 F1 T1 j€J

-p u-p v e

l

k

d |v + i | 1

k

j

5 77

' ' jej

|u +i | 1

1

j

S 77 '

'

JGJ

I |A

we again see that

|v + i |

j1

1

j

dudv

0

5-p 8-p d-p 1 k |v+i| |u+i| |u+v+i|

By applying Lemma 4.1

| u +|A

j k l

-p

I k

dudv.

the integral

(4.21) on

sector W converges if and only if one of four sets (4.22) of conditions is true. This completes the proof of Theorem 4.1. ■

Now we formulate analogous criteria to establish convergence of the H-integral in (2.2) for complex variables x and y. 38

Corollary 4.5. Let the contour L run from c-ioo to c+ioo, and L,_ run s t from C-ioo to C+ioo in the complex s- and t-planes respectively. Let there also be at least

one pair

Cj,kJ, j,k e I, such that

a Aj k

A.a * 0. Then the U-integral

in (2.2) converges for x,y

j k

if there exists with I),

some subset

I of the set

for which the following

1) | A argix)

- a arg(y)| < up /2

k

k

3) p < -1 for

may be empty or

four conditions for

2) |A arg(x) - a arg(y)I = np /2 1

I (it

'

€ C if

are true

and only coincide

simultaneously

k e I .

for

k e I\I .

k

0

k € I\I . 0

^k

4) Among lines lines

(In

L , k e the

I\I , there

other

case

are no pairs

additional

of

condition

neighboring 8 < -2

is

required).

Geometrically

this

Consequence

means

that

the H-integral (2.2)

converges for x,y G C, if all p > 0 , k e l , and coordinates of point the [(argix), part

of

|A argix) 1

arg(y)] of the [arg(x),arg(y)]-plane lie in the internal the polygon

-

k

formed

a arg(y) I < np /2. k

'

by

the intersection

of

all stripes

If they lie on the boundary

then the

k

condition p < -1 also required. And if it coincides with some vertex of polygon then for convergence of the H-integral

(2.2) we also need that

8 < -2. Finally, we describe process

of establishment

of convergence or

divergence of the H-integral (2.2) for x,y 6 IR+.

Step y

1.

Check m ^ 2 and exist Jyj,k € I such that a A

j k

- A a

j k

* 0.

Otherwise the H-integral (2.2) diverges.

Step

2. By the formula (4.9)and (4.15) calculate F(u,v) and p ,k e I,

respectively. 39

If there exists at least one k, k € I, such that p

< 0, then the

H-integral diverges.

If all p

> 0, then the H-integral converges.

Step

3. Let all p ^ 0. Then find the subset I of set I such that ^ *k o p > 0 for k e I and p = 0 for k € I\I . 'k o ^k m o Step 4. From (4.14) calculate a. for j € I. In accordance with Definition 4.1 find at least one pair (k,l), k,l e

I\I , such that L 0

and L

k

are two neighboring lines.

If not exist, then conditions of the convergence of the H-integral (2.2) holds kind

p

< -1 for k € I\I . k

0

Otherwise, we obtain the conditions p < -1 for k € I\I and 8 < -2, ^k o where 8 is defined by (4.18).

Now we consider typical examples for illustration of this criteria. First we again examine the integral (4.1).

A 1 4.1.

IT i Example

Qr

40 = r(a+s)r(/3+s) ! e(s,t) . r(*+s+t)

Here

(a ,A ) = (1,0),

(a ,A ) = (1,0), (a ,A ) = (1,1). Hence 2

2

3

3

Step

1. Here m = 2 and det

I v

v

Step 2. Step

Pj

3.

= p 2 = 0,

I = {3>, o

8

= m, 1

I\I

o

1

0 '

det

= 1*0. ^0

\-

1

= {1,2}. 8

= <2>,

J

2

r

=

p 3 = 2.

= Re(a)+c-l/2,

J

a

Al ]

= Re(0)+C-l/2, J 3

2

40

= <3>.

8

= -[Re(y)+c+C-l/2].

p

=5+6 2

*1

p

=6+5

^3

Step

Hence L

p

= 5 +5

2

= Re(a-y)-C,

1 3

= Re(a+/3)-c-C-l.

1 2

4. a ^

= Re(|3-y)-c, 3

= 1, a 1

= 0, a 2

= {u = 0>, L 1

= -l//~2. 3

= {v = 0} 2

are two neighboring lines. Consequently,

we obtain the convergence conditions

Re(/3-9-)-c < -1,

Re(a-?)-C < -1,

6 = Re(a+/3-?)-— < -2.

Note that if a = |3 = y = 0, i.e , e(s,t)

r(s)r(s)

B(s,t), where

T(s+t) B(s,t) is beta-function inequality

allows

us

(see A.Erdelyi et al. (1953)), then the last

to

conclude

that

the

corresponding

diverges for all c, C e R.

Example 4.2. 6(s,t)

_ ru+s+t)r(/3+s-t) r(*+s)r(7j+t)

1 Step

ll

1. Here m = 2 and det

=-2*0. 1

Step

2.

p = p = 0 ,

Step

3.

I = {3,4}, o

*i

6

*2

-1

p =Kp = 2

^3

4

I\I = {1,2>. o

= Re(a)+c+C- — 1

5

5

2

= -[Re(*)+c-— ], 3

= Re(/3)+c-C-— ,

2

5

2

4

41

2

= -[Re(T>)+C-— ]. 2

H-integral

J

= {1>, 1

p

=6+6+5 2

*1

3

J

= {3>,

= Re ((3-7-7)) - 2 C + — ,

= 5+6+5

^2

a

= 1, a 3

=0.

= { u+v = 0 } and

l

lines.

C o n s e q u e n t l y , we o b t a i n t h e c o n v e r g e n c e Re(/3-?-T))-2C+—

Example 4 . 3 .

< -1,

Re(a-y-T))+ —

ri

n

0 p = p = 0 ,

I = J

5

3

^4

= {3,4,5,6},

I\I

= {1,2},

J

= J 3

2

5

tfe(y)-C-^-

'l

^6

= {1,2}.

= {3,5},

=p

3

= Re{v)+c+C-

1

+5+5 4

5

— ,

=tfe(/3)-c-— ,

5

= -[/te(y ) - C - — ]. 6

2

1 1

42

= {4,6}. 6

6

= Re(/3+y-/3 - y 6

= J 4

5c = -[Re(R ) - c - - i - ] , =5+5

2

J

5

5

p

^5

0

= Re(a)+c+C-—,

54 =

= - 1 * 0 .

-1

0

1

< -1.

p = p = p = p = 4 .

2

*1

conditions

r(a+s+t)r(y+s+t)r(/3-s)r(y-t) r o i +s)r(* i +t)

9(s,t)

1. Here m = 2 and d e t

J

2

4

= { u - v = 0} a r e n o t n e i g h b o r i n g

Step 3.

= Re(a-r-r?)+ — . 4

1 3

= -l//~2,

= l//~2,

S t e p 2.

= {4>, 4

p

2

Hence two l i n e s L

Step

J

3

2

L

= {2},

4

S t e p 4. a

a

J

2

)-2(c+C).

1

2

Step

4. a

a

= -l//~2.

2

Hence L

L

1

= {u+v = 0}. 2

Consequently, we obtain Jte(0+r-0 -y )-2(c+C) < -1. l

Note that if here 0 = y

l

then provided the last condition holds

we obtain Appell series F . Now we give the Mellin-Barnes type integral representations for 15 out of 34 Horn's functions of

two variables (see

A.Erdelyi et al. (1953)) with the convergence conditions. We write

H(x,y) =

9(s,t)x y dsdt, (2711)"

L

L 2

where t h e c o n t o u r s L

and L 1

1

r u n from c-ioo t o c+ioo, and C-ioo t o C+ioo i n 2

the s- and t- complex planes respectively. We also assume that these contours satisfy the poles separation supposition for the kernel 0(s,t) (in each situation). For

convenience

we

introduce

the

Slater's

notation

(L.J.Slater

(1966), O.I.Marichev (1983)) Hot ). ..T(a ) 1

(4.33)

ro ).. .ro )

8 ,...,0 1

p_

q

i

q

It is to be noted that other functions from Horn's list cannot be represented

by

analogous

integrals

with

vertical

possible only for the other cases of the contours.

43

contours.

It

is

TABLE 4 . 1

e(s,t)

H(x,y)

a - s - t , £ - s , / 3 ' - t , s, t

" a./s.P* 1

F (a.0,0';y;-x,-y)

r

r

»

y-s-t

*

\arg{x) | < 7i, |arg(y) | < n

' 2

r

a - s - t , £ - s , / 3 ' - t , s, t

*,?,&' F2(a,{3,/r;y,y';-x, -y) y,y'

r

'

y-s, y ' - t

|arg(x) | < 7T, | a r g ( y ) | < TT, | a r g ( x ) - a r g ( y ) | < TT

a , a ' ,/3,/T" 3

r

f* a - s , / 3 - s , a ' - t , / 3 ' - t , s, t F ( a , a ' ; 0 , £ ' ;y;--x,-y)

r

*

\arg(x)\

" a,|3 1 4

r

< TT, \arg{y)\


a-s-t,/3-s-t,s,t 1 F(a,P;y,y';-x,-y)

y>y'J

'

y-s- t

r

4

»

y-s, y ' - t

x,y € [R+, 2c+2C-/te(y+y' ) + l < 0

' a-2s-t,/3-t,s, t ]

" «,£' 5

#3(a,|3;y;-x,-y)

r

r

*.

»

y-s- t

| a r g ( y ) | < TT, | a r g ( x ) |

| a r g ( x ) - 2 a r g ( y ) | < TT,

1a r g ( x ) - a r g ( y ) | 44

<

2TT

< 3TT/2,

" a,0 6

r

-.

" //

.

y,<5

4

r

(a,0;r,5;-x,-y)

a-2s-t,0-s,s,t y-s,

s-t

-, J

|arg(y)|

<

n/2,

|arg(x)-arg(y)|

" a,01 7

r

*J

' a-s-t,0-s,s,t $

r

(a,0;y;-x,-y)

r

' 0,0'1

»

;y;-x,-y)

r

V

r

' &1 _7J

< TT/2, | a r g ( y ) |

*

y-s-t |arg(x)|

y € R+f

" a,0 10

r

. 7,7'

1

" 0-s,s,t "

r

3(0;y;-x,-y)-

1

< TT/2,

c-/*e(y) + l < 0

" a-s-t,0-s,s,t

U1(a>0;r,7,;-x>-y)

r

y-s,

y'-t

|arg(y)|

" »

< ir/2,

| a r g ( x ) - - a r g ( y ) | < 7t/2

45

< n/2

y-s-t

|arg(x)|

9

< 7i, \arg(y)\

" 0-s,0'-t,s,t <M0,fr

"

y-s-t |arg(x)|

8

< TT/2

< TT/2

r 11

_

OL

* (a,y>y';-x,-y)

r

r

y,y'J

a-s-t,s,t y'-t *

y-s,

x,y € R+, 2c+2C-/te(y+y')+l < 0

a - s , / 3 - s , a ' - t , s, t

a, a' ,/3 12

2 ( a , a ' ,/3;y;-x,-y)

r

7

r

\argM

H (a, /3; y; -x, -y)

r

| < irf

a-s,/3-s,s,t

CL,(3

13

r

7

'

y-s-t

|arg(y) | < TT/2

-i

y-s-t

-1 | a r g ( x ) | < 7T,

y € IR+,

c+2C-/te(y)+l < 0

' a-2s-t,s,t 1

a 14

r

H (a;y;-x,-y)

r

6

7

y-s-t

*

\arg{x) | < 7i, \arg(y) \ < n/2 \arg(x)-2arg(y)|

1" a - 2 s - t , s, t

a 15

r

. *>6.

< n

H (a;y,5;-x,-y)

r

y-s,5-t

>

x,y € R+, 2c+2C-Jte(y+5)+l < 0

46

§5. Simple contiguous relations for the H-function of two variables In this section we establish

certain

constant coefficients for the H-function

contiguous

relations with

(2.2). Here we simply write

H(x,y) if all the parameters of the H-function of two variables are as in

(2.2), H[a +1] for the contiguous H-function

which

a

is

replaced

unchanged. Similarly

by

a+1,

but

all

other

of two variables in parameters

are

left

we introduce notations H[a±l], k = 1,2,. 1,2,..,m, . ,m, and k

H[p.±l], j = 1,2,..., n. Equalities involving H(x,y), H[a±l],

withJ constant c o e f f i c i e n t s

are c a l l e d contiguous r e l a t i o nks .

H[0.±1],

But

j those

with constant coefficients are called contiguous relations. But those contiguous relations involving only H(x,y), H[a +1], H[/3 -1] are called simple.

The

general

theorem

of

contiguous

relations

given

in

this

section was obtained by R.G.Buschman (1990) and here we little intensify it Now in the same way as for the H-function (2.2), we write e[ct e[ak+l], +1], k = l,2,...,m, and e[|3-l], j = l,2,...,n, for the contiguous function 6(s,t)

(2.3)

in

j

which

a

is

replaced

by

a+1

k

respectively.

and

0 (3

k

by

j

Applying the equalities r ( ll++cat + a s+ A t ) = (a + a s+ A t ) T ( a +a s+A t ) , k

■"

k

k

„

1

_

rr(/s ( 0 -i+b -1+b S s+B + B t) j j

j j

k

k

k k

k

k

0 -1+b s+B t J

J

J

r(p r(0 +b +b s+B s+B t) t)

j j

j j

Jj

j j

we we obtain obtain m m

nn

ii- m m

nn

-. -.

y a eta +i] + y c.et^.-i] = y ^kak + y €.0. e(S,t) y a e[a +i] + y k k= = ll

(5.1) (5.1)

rrm + r[ r m ) \ \ + [ ) \ \ L v k= l L I k= 1

for any 7)

'k

, k = 1,2

^kak + y

•k= l l "- k=

j= j= l l

+ +

y

€.e[/3.-i] =

j=l j=l

^ ^ . e(s,t) -I -I

n ^ ( m n ^i )n € > . ^s + ( \m 7)kAk + n \ C.B.^ t i e(s,t), ) €>. s + \ 7>kAk + \ C.B. t e(s,t), j= l > ^ k= l j= l ' J j= 1 J V k= 1 j= 1 ) J m and ? , j = 1,2 ^Jj

47

n.

0-1, j

Consequently, this allows us to formulate the following theorem.

Theorem 5.1. Let

m+n numbers

T? , k = 1,2, . . , m and £ , j = 1,2,. . . , n, k

satisfy

the next two

j

equations > i)k k a + ) £ b = 0,

f

\ L

L J J

k=l

j=l

(5.2) m

n

> 7) A k=l

Then the following

simply

C B

= 0.

j=l

contiguous

m

(5.3)

+>

relation

n

is

i- m

n

^T, k H[a k+ l] + £ W - l ] = £\«k k=l

true:

+

E ?A

H(x,y).

j=l

j=l

This theorem is simple to prove. Here it is sufficient to note that T) , £ if constant coefficients 7)

satisfy ((5.2), then the second addend in

the right part of (5.1) is equal to zero.

Theorem 5.1. shows that all simply contiguous relations are defined by m+n roots (T) , 7) ) of system (5.2) of two equations with respect to the known a ,A ,b ,B . Evidently, the rank of this system is equal to 2 y H "

k' k' j* j

(otherwise

the H-integral

diverges).

relations for the H-function

Hence

there

exist

m+n-2

basic

(2.2) of two variables, which define all

other simple contiguous relations. We describe it by the following two examples. Example 5.1. e(s,t) =

Hex +s+2t)T(a -/~3s+t)r(a +s) .

r(^-t) Here in accordance with p

= 2+/~3,

p

=2,

p

(4.15) in Theorem 4.1 we have p

=3.

Hence

converges at least for x,y € IR+.

48

the

corresponding

= 2+2vr~3 H-integral

Further the system (5.2) has definite form

7) -V~3 7} + 7) = 0, 1

(5.4)

2

3

27) + T) - £ 1

2

=0.

[T) , TJ ,-TJ +Vr1^7} , 2TJ +TJ ]

It is not difficult to see that 7)

^1

for

any

7) ,

is the root of the last system of equations with respect to four

unknowns

[T? ,TJ ,TJ ,£ ]. Hence

from

(5.3)

we

obtain

general

simple

contiguous relation

77 H[a

+1]

+ 7) H[a

+1]

+

(/~3T) -TJ )H[a

+1]

+

(2T? +7) )H[0

-1]

[a T? + a i) + a (V~3T) -TJ ) + 0 (2TJ +TJ )]H(x,y). ,J 11 2 2 3 2 1 1 1 2

Evidently, (T) ,7) )-space

two and

vectors

they

[O^.v'-3,1]

[1,0,-1,2]

and

contiguous

relations

(1,0)

define

for

and

two

(0,1)

following

respectively. this

are

roots

the of

Consequently,

H-function

can

basis

system all

simple

by

linear

obtained

combinations from two basis relations

H[a +1]-H[a +1] + 2H[0 +1] = (a -a +2/3 )H(x,y),

H[a +1] + /~3H[a +1] + H[0 -1] = (a +/l?a +0 )H(x,y). 2

Example 5 . 2 .

3

G(s,t) =

1

2

3

1

T(a + / ~ 2 s + t ) r ( a - s + t ) T ( a - s ) T ( a - t ) —

r o -s-t)r(js -s)

49

of

(5.4)

= 3V~2,

Here after some calculations we obtain p

p

= V~2,

p

= 2,

'l *2 ^3 p = V~~2,p = 2+V~^2,p = 2. Hence the corresponding H-integral converges

4

5

6

at least for x,y e IR+. From (5.2) we obtain the next system

( V ^ T ) -1) -n -£ -£ = o, 1 I V

With

respect ^

to

six

2

3 ^1 ^2

TJ +TJ -7) " 5 = 0 . 1 2 4 ^1

unknowns

[TI ,TI ,TJ ,TJ ,6 ,6 ] we V 2 3 4 ^1 ^2

If) ,V >T) >T) >i) +T) ~V , (/^-1)TI -2T) -7) +7) ]

for

any

have

the

17 ,7) ,7) ,7) .

root Hence

V 2 3 '4 1 2 4' 1 2 '3 4 * V 2 '3 4 from (5.3) we obtain the following general contiguous relation

77 H[a +1] + 7) H[a +1] + 7? H[a +1] + 17 H[a +1] +

(^^-irjjHt^-l] +

[(Vr^-l)7)i-27)2-773+774]H[^2-l]

= {a 7) +a T) +a 7) +a T) + B (TJ +T) -T) ) 1 1 2 2 3 '3 4 '4 1 1 2 4

+ ^2[(/^-l)77i-2772-7)3+774]>H(x,y).

If vector (7) ,7) ,TJ ,7) ) is equal to (1,0,0,0), (0,1,0,0), (0,0,0,1),

then

we

obtain

four

following

basis

simple

(0,0,1,0), contiguous

relations respectively Htc^ + 1] + Ht^-1] + ( T T 2 - 1 ) H [ 3 2 - 1 ] = [e^+0 +(iT2-l)0 ]H[x,y],

H[a2+1] + H[^-1]-2H[02-1] = ( a ^ - 2 0 )H[x,y],

H[a 3 +1]-H0 2 -1] = (a3-^2)H[x,y],

H[a +1] - H[p -1] + H[0 -1] = (a -0 +0 )H[x,y]. 4

1

2

4

50

1 2

§6. Main properties for the H-function In this section we assume that the H-integral in (2.2) converges. By Hartogs

theorem

H-function

it

is not

difficult

to obtain

that

the general

(2.2) is analytic at least on the region described as

follows: r

m

n

|arg(x)| < \ u

jj==i i

j=i

m

n

j =i

j=i

|arg(y)| < L

Here, as usually, we denote (a,a,A) = (a ,a ,A ),.

(a ,a ,A ), m m m

(0,b,B)n = ( f ^ . b ^ ) , ,

O

i

m

l

l

n

,b ,B ). n n

It is not difficult to get two evident properties H [x,y; (a,a,A) m-l

, (/3 ,b ,B ); 0,b,B) ; L ,L.] n

n

n

n

S

t

H [x.y; Coe.a.A)^; O . b . B ) ^ ; L ^ ] .

|H [x,y; (a,a,A)m,(0,b,B)n; L s,LJ| t J dxdy ^ = -^- H [x,y; (a,a,A) , (0,-1,0), (0,0,-1); xy m (0,b,B) ,(1,-1,0), (1,0,-1); L ,LJ. n s t Further we will consider the case of the H-function where the contours L and L^ are vertical lines L and L_ with real parts c and C s t c C respectively, i.e. 51

L

= {s, Re(s) = c>, L c = {t, Re(t) = C>.

Suppose that the corresponding H-integral H[x,y; (a,a,A) ; O,b,B) ; L Lc,L ,Lc]] (6.1) = -- i i

— (27ri)2J

L

J J

r

C

— (27ii)2 JJ

J J

t 9(s eCs,Dx'V^sdt =— —)t)x"V dsdt =

Re(t)=C

L

e(s,t)x"ssy"ttdsdt

Re(s) =c

converges. By suitable replacements of s and t in the last integral it is not difficult to obtain that 1. The change property xpyqH[x,y; (a,a,A) ;(0,b,B) ;(|3,b,B) m

n

; L ,Ln] c C/

(6.2) = H[x,y; (a+pa+qA,a,A) ;(0+pb+qB,b,B) ;(0+pb+qB,b,B) ; ; L_ _ ,, L ]; ]; 2. The strain property H[x p ,y q ; (a,a,A)m; (0,b,B)n; 1-^,1^] (6.3) = =

X T^H[X'y;

(a>a/P>A/<*V^b/P>B/q)n > Lc 'LCq ]; (a>a/P>A/^m^^b/P>B/q)n ;L C pP,LCq];

3. The linear property H[x,y; (a,a,A) ;(0,b,B) ; L ,L ] (6.4) k p h = I| A | H [x y ,x q y ;U,ka+ PrA,qa+hA) .;(/3,kb+ ;L ,,L [xV,xV'.Ca,ka+pA,qa+hA) ;L LCnn ], ], (/3,kb+pB,qb+hB) I J J -i J P-B,qb+hB) - I n c m

J

II

where A = kh-pq * 0

J

c

r

-i

m

r-

i

= (hc-qC)/A, C = (-pc+kC)/A.

n

c

C

Here in a l l the

above formulas D a k h are real constants , and pq * 0. Now

we

p^tabli^h

a^vmntot i p

p«?t imat i nnc;

for -far

thf

H-fnnnMnn

(6.1). (6 1) From Definition 2.1 2 1 we have (6.5)

|x" |x"ss|| = |exp|-sriog|x|+iar^(x)l||= |expj-s[~log|x|+iar£(x)lj| = eexp[arg(x)7^(s)j|x|' x p f a r g ( x ) 7 f l 2 ( s ) j | x | ' CC. .

52

Similarly, |yt_t||== exp[arg(y)I/n(t)l exp[arg(y)I/n(t)l |y|" |y|"CC.. |y

(6.6)

Since the H-integral

in (6.1) converges, then as is known from

§3 — § 4 we have

((6.7) 6.7)

M = JJ_ f \ f|e(s,t)|exp[ar*(x)I«(s) L(s,t)|exp[ar*(x)I«(s) + arg(y)Imit)1 |dsdt | < +<*, 4TI 2 J 4TT L

J

L

J

CLc

Denoting the H-function (6.1) by simple symbol H(x,y) from (6.5)-(6.7) we obtain |H(x,y)| * M |x|'C|y|"C for x,y € C.

(6.8)

In accordance with Definition 2.1, the contours L and L„ do not c C intersect any singularities of the integrand e(s,t), i.e. intersect any singularities of the integrand e(s,t), i.e. ReU + a c + A C) * 0, -1, -2, ... .. . j

for j = 1,2

j j

Jj

m. Hence for all j = 1,2

m there exists e > 0 such

that /te(a.+a s + A.t) * 0, -1, -2,... for s,t, such e+c and -e+C < Re it) function

9(s,t)

that -e+c < Re{s) ite(s) <

< e+C. This means that for these s and t the

(2.3) defined

by

the ratio

of

two products of

gamma-functions is analytic. Consequently it is not difficult to obtain that H(x,y) = —

9(s,t)x' e(s,t)x'ssyy_tclsdt, \isdt,

2

(27ri) J J L

C Lc l

l

where (6.9)

-e+c -e+c < c < c e +c, +c, l

-e+C -e+C < C < e+C. l

Applying the estimation (6.8) we have -c -l |H(x,y)| * M|x| 1|l| |l| .

53

for any c , C , satisfying the conditions (6.9). The last estimation can be written symbolically as follows |H(x,y)| *M|x|- C ± G |y|- C ± e

(6.10)

where e is a sufficiently small real positive numbers.

§7. Let

The double Mellin transform

the function f(x,y) be defined

in R? =

(0,+oo)x(0,+a>). Then (0,+oo)x(0,+co).

classical double Mellin transform of function f(x,y) in point (s,t) € C cl ssiced thu lelMowlng inansfor 00

mjf(*.y);s,t^

(7.1) (7.1)

00

p p p

f(x,y)xss 1yyt f(x,y)x

= f*(s,t) f*(s,t) = = =

X

dxdy. *dxdy.

0 0

This transform plays important role in the construction of the theory of integral transforms and convolutions. Here we give main properties of this transform. Theorem Theorem

Let Let

7.1. 7.1.

IIIL cgla.1

III

L lie

nitcgia.1 that that (7.2)

in

Liic r ±gtii c

r lgtl

aloo

the

L

pixl

L

parL

< Res))

l

l

Let

function function function

l

function

continusus continuous continusus COilVSlgcS

I OL

d.1 1

au£>oiutciy

converges

ior

a±± s,t,

w .1 ) au£>U ILL t c ly

w .1)

< E

< Reit)

on on

R+ and R+ and

s,t,

the the

SUCH

such

< E E.

2 2

2

f(x,y) be reprenented

f(x,y) = — l — —

(7.3)

c

and

l

2

f(x,y) be f(x.y) be

by the

ffrmula

F(s,t)x"sy" F(s,t)x"sy_ttdsdt,

2 2

(27ii) (2ni) J\ J L

where

En L^ L

respeceively respeceively respeceively

and

L

c

mean

(heee (heee

f*(s,t) = F(s,t) for

e e

vercical vercical

< < c c < < E E , , e e all

s,t such

C

lnnes

L

c wihh

<
54

Reis) Re{s)

Then Then

real

parss

C

and

3Jt 3Jt
= c, Rei))

= C.

c

Theorem 7.2. Let theee

srrsps

the

1. Function Function Function

next

s,t belong

to

conditions it ions

hold

strips

defined

by

(7.2) and

in

s,t beoong

to

F(s,t) F(s,t) is is analytic. analytic. analytic.

2. The following 2.

integral

converges

(7.4)

|F(s,t)|dsdt << +O0 +«, L

C Lc l

for

all

(7.5)

c , C , such l

l

t that hat

l

e < c < E ,, I i

ll

from

strips

(7.3) it it

e
ll

2

3. |F(s,t)| 3. |F(s,t)| -> 0 for for

Then

the

2

11

2

||s|, s | , |t| | t | -> -+ --Hoo. Hoo.

foiiows follows

that

f*(s,t) f (s,t) = F(s,t), where

(7.2).

Readers interested in the proof of these theorems can find them in I.S.Reed (1944). Here we note that integral (7.1) absolutely converges. In fact, since function F(s,t) is analytic in the strips (7.2) then from (7.3), (7.4) we obtain f(x,y) f(x,y) = — — F(s,t)x"sy"tdsdt, (27Ti)2 J J JJ

(7.6)

L

C

for

L

ll

c

c , C 1

|| ff (( x x ,, y y )) ||

1

s a t i s f y i n g (7.5). Therefore -C -- cc -u 1 < < M Mx x * ' y y 1 where where

M= J J_

ll

for

all

|F(s,t)dsdt|

4TT 2 J J L

C

JJ L

ll

c

ll

55

x > 0,

<+oo. <+c».

y > 0

we h a v e

Hence for sufficiently small e we obtain -c -c -C -e |f(x,y)| < M x 1 y 1

for x > 1, y > 1,

-c +e -C -e |f(x,y)| < M x 1 y *

for x < 1, y > 1,

-c -c -C +e |f(x,y)| < M x * y *

for x > 1, y < 1,

-c +e -C +e |f(x,y)| < M x l y l

for x < 1, y < 1.

—

The last inequalities allow us to conclude that f(x,y)x R6(s)-1yR6(t) 1 € L(R+), i.e. the integral in (7.1) absolutely converges. Theorem 7.3. Let 9(s,t), defined

by (2.3), satisfy

the

following

conditions Re(a +a c + A C) * 0, -1, -2, ... j

(7.7)

j

|e(s,t) dsdtl < +oo. L

Then for

j

the H-function

C Lc

H(x,y)

e(s,t)x"sy"tdsdt

H(x,y) = 2

(2Tui)

L

we obtain

the next

C Lc

equality

3JI JH(x,y); s.tj = 9(s,t) for

|/te(s)-c| < e,

positive

|Re(t)-C| < e, where

e

is

a

sufficiently

small

number.

Proof. Since Re{a

+ a c + A.C) * 0, -1, -2, ..., then there exists 56

Re{a

+ a s + A t) * 0, -1, -2, ... , j

j

j

(7.8) for Consequently

the

|/te(s)-c| < e , |/te(t)-C| < e .

function

6(s,t)

(2.3)

is

analytic

in

the

strips

described by (7.8).

Further from (7.8) and Theorem 4.1 it is not difficult to observe that

|0(s,t) dsdt| < +oo, L

C

L

l

c

l

where |c-c | < e , | C-C | < e , e

is a sufficiently small positive num­

ber.

applying

If

set e = min(e ,e )

and

Theorem 7.2, we obtain the

Theorem 7.3. ■

§8. Series representations for the H-function of two variables

In a general case the problem of representing the general H-function of two variables in terms of the residue sum in the integrand function poles, which is very important for the theory and applications, involves great difficulties, which cannot be overcome yet. An attempt to use the multidimensional residue theory, given in an account of L.A.Aizenberg and A.P.Yuzakov (1979), A.K.Cih (1988),

to the

double

It

H-integral

(2.2) has

not

given

an

essential

result.

is

connected with complicated behaviour of the integrand function 0(s,t) (2.3), when s,t —> oo, s,t € C.

However, in a great number of particular cases the H-function (2.2) can be represented by the sum of double power series of hypergeometric 57

type. In this section we shall show that if the integrand function 9(s,t) (2.3) has the next form

m

m

o

2

6(s,t) = T~Tr(a -a s-A t)rTr(a'+a's)rTr(a ,, +A't) > : J J J j=i' J J j=i' j j j=i

(8.1)

where

m

1

a

£ 0, A £ 0

j

for

j=l,...,m;

j

a' > 0,

o

j=l,...,m;

j

^

A' > 0,

l

j

j = l,...,m , then the corresponding H-function can be represented by the sum

of several

double

Theorem 8.1 below).

series

Here

under

we note

some

that

additional

the situation

conditions

(see

a. ^ 0, A. ^ 0,

a' < 0, A' < 0 is leaded to (8.1) by the replacement x by x

and y by

y"1 in the H-integral (2.2).

Here we also present several examples, which give solution of this problem in the other situations (unlike 8.1). The method

to establish series representations of the H-function

used twice in Slater's theorem, which is based on the residue theory of the

gamma-function.

This

well-known

theorem

appeared

in

L.J.Slater

(1966) and later it considered in detail in O.I.Marichev (1983).

Now denote

(a,a,A) m

= (a ,a ,A ),...,(a ,a ,A ). i

l

l

(0,b,B) n = ^ . b ,B ),

where a , 0

m -

m

m

(/3_,b_,B_), n

n

n

<=C, a , A , b , B € IR such that 8 +b p+B q * 0,-1-2, . . .

for all j = 1,2,...,n; p,q = 0,1,2,.. type series

58

Then the double hypergeometric

in

^ (8.2)

F [x,y; (a,a,A) ; (0,b,B) ] = m n

1 Tr(a +a p+A q)

xPF y 4

^ 0 TTrOj^p^q)

p! q!

> /

_ -^-1 n

j j

q

j

P - - j-l

converges absolutely for x,y € C, if

m

(8.3)

n

m X = 2

X = V a.- V b -1 < 0, j=i

j=i

/ A" / j=i

n B _1 <

°'

j=i

If A > 0, A > 0, then series (8.2) diverges whenever x * 0 and y * 0. 1

^

2

This fact follows immediately from Horn's method for double series (see J.Horn (1889, 1931)) and the asymptotic behaviour of the gamma-function. If all these coefficients a , b , A , B j

j

j

are equal to 0 or 1, then the j

corresponding series is different from Kampe de Feriet series only by some constant. This known series will be studied in §14.

Now we consider the following particular case of the H-function (2.2) in the next form

H[x,y; (a,-a,-A) , (a', a' , 0) , ( a " , 0, A' ) ; (/3,b,B) ; L , L. ] m m m n s t o 1 2 (8.4) m

(2iri)2

m m o 1 2 Y~\ r(a.-a -s-A.t)"T~[ r(a'+a's)"TT Ha'+A't) } r J J j j J J j=l j=l -s -t . .. n_ J = l J x y dsdt. _ n ^ b B ) Ll L V t j t ~s j=l

59

For (8.4) there is the next theorem.

Theorem 8.1. Let

(a

£ 0, A

£ 0

for j = 1,...,m ;

(8.5) a' > 0

Let all be

left

simple,

poles i.e.,

simultaneously

(8.6)

a

(8.7)

j

Let

i

J

s = -p and *

1

poles

the contour

+ a 2

finally,

in the right

of equality

j

s = -q, M 2

a'.'+ A', t = -q, J

2

function

2

respect

(respectively

L

f

m

part

of (8.4)

cannot be

satisfied

j=i

j

1 2

2

part

of

1

2

1

j = 1,

m 2

(8.4) satisfy

the contours all

L

the

and L ,

the poles

(respectively

of conditions

j=i

m 2

o

n

j=i

j=i

j=i

60

J J,J = l,...,m ;

i

j * j •

m when t € L

two sets

J

of

T(a' '4-A't), j = 1, . . . ,m ,) from

(8.8) m

2

to

m

j=i

i

L ) must separate

(respectively

the following

j * JJ ;

J

in the right with

of f(a -a s-A t), j = l

Let,

function

two sets

j

supposition

T(a' +a' s), j = 1, . . . ,m poles

a

J

i

the integrand

separation

and A' > 0 for j = 1, ,m

p,q = 0,1,2,...

a " + A' t = -p and J

i.e.,

j

of the integrand the following

for

+ a 1

for j = 1, . . . , m

be also

s € L ) .

valid

the

m o

m

a

1

+

E j E a j "Z j=i

j=i

m o

|b| > |arg(x)|

j *

j=i

m

1

A +

E i Z AJ " Z J=I

J=I

|B| >

J « i a r g ( y ) i,

J=I

(8.9) m +n o

m

(a det

j

sgn(m +1/2-J) + A \ a' + a \

j

A

A'

A k

j=i

m

A 1 j=l

k

j=l

f arg(x) arg(y) ) det

a k

a

m +r 0

Then the H-function series

, for

A

all

k

= b , A r

m +r 0

= B , r = 1,...,n. r

(8.4) can be represented

of form (8.3) as

k = 1 , . . . , m +n, o

by the sum of

double

follows

H[x,y; (a,-a,-A) ,(a',a',0) ,(a",0,A') ; (0,b,B) ; L ,L.] m m m n s t 0

m

1

2

m 1

a'/a' a'VA'

2

k

k

■II-

h

h

y

1/a'

„

x

F

k

1/A' h , -y

(8.10) a a ^ k ^ h . a A a+ — a + A ,— , — a' A' a' A' k

h

a

h

a

A

» n

k

k , a a - — a ,- —

m -l l

h

a

A A

a

,

_

0- Jib --*-B ,-*- ,- *.

- — A ,0,- — A' A' h

_ ,0

a' k

h

61

A' h

a' k

A' h

where k , a a - — a ,- —

_ , 0 m -l 1

a

1

>

a - — 1

a

n

a ,- —

,

,0

k-l

1

a

k

a k

a

a a k k-l - — a ,, 0 , k-l a a

a k

(8.11)

k

k

a a k , k+l ,0 - — a ,k+l , k+l a a k

x

k

a k - — a ,m , m l a l k

m 1

, a k

and so on.

Proof. At first we note that according to Theorem 3.1

conditions

(8.9) are required for the convergence of the double integral in the right part of

(8.4).

But

the two

inequalities

in

(8.8) provide

the

absolute convergence of all the double series which are involved in the right part

of

(8.10)

(see also

(8.2) and

(8.3)).

Hence

the double

integral in (8.4) can be evaluated as follows. Transform integral (8.4) in the next form

2

1 27i i

TTr(a' '+A't) y_tdt 1 j =i

(8.12)

m

1

2ni

r 1

m o 1 ~TTr(a -a s-A t)"TTr(a'+a,s) J=i J=I

x ds.

r r n p +b S+B U j=i

Denoting the inside integral in (8.12) by D , we see that its integrand s = -(a'+p)/a>, J J j = l,...,m , p = 1,2,..., and the right set s = (a.-A.t+p)/a., t € L .

function

has

two

sets

of

poles:

i

the

left

set

J

62

J

j

*-

Moreover, in accordance with (8.6), we get that the left poles are simple. Hence from the ( residue \ theory (-l) p of the gamma-function for s = ~(a'+p)/a' we get res T(a'+a's) = , p = 1,2,... . j

j

v

J

)

J

a>

p!

j

Now applying Slater's theorem to the inside integral in (8.12) we obtain that D is equal to the sum of residues of all the left poles, i.e., Ill

m i r—,

> ,

«. /<x a /a

,

r—.

r

U.

CX

\

^

a'

a'

'

r r r « + -± a -A t+ -i P j=i

V

j=l m l—1>[

T^rV j=i j*k

Substituting D

v

»

, k

, j

,

T V "7 a

a

J

a'

J

k

(8.13) ]

p

{-IV

TT~

;

a' k

x

p/a'

k

^

(8.13) in (8.12) and again applying Slater's theorem

we obtain representation (8.10). Thus Theorem 8.1 is proved.

■

Example 8.1. Let us

H[x,y; (<x,l); (0,1); L ,L ] S' t

l

, ..2 (2TTI)

r(a+s)TO+t)x s y \isdt. L.t L s

Then the convergence region of the last integral is described as follows |arg(x)| < TT/2

and

|arg(y)| < TT/2.

In this case according to (8.10) we get

(3 -x -y H[x,y;(a,l);(/3,l);Ls,Lt] = xV^T-x.-y; - ;-l = x°V e e J . 63

Remark 8.1. The above repeated method, which is used to establish representation (8.10), in general case transform the general H-function of two variables (2.2) in terms of the sum of several incomplete double series of the next type

xx vVyVL^ Ss» (xV) (xV)p (x'V (x^y ))q,,

(8.14) (8.14)

p

P

q

q

(p,q)€A

where A is some subset of the set IN2 = IN x IN , where IN denotes the set of

non-negative

integers.

This

situation

appears,

oor example, in

o f u d n o n - the next H-funers.n H[x,y;(a,1,1),(0,1,0),(0,0,1); (8.15) (8.15)

L ,L.] S L s L

♦ + ico i o o +ioo +i»

= — 121 = — {2niV2 {27ii)

Remark 8 . 2 .

JJ J J -ioo -ioo -ioo -ioo

t s r(a+s+t)r(s)r(t)x"y dsdt, r(a+s+t)r(s)r(t)x" y' t dsdt,

Theorem 8 . 1 shows t h a t

(2.3)

can be r e p r e s e n t e d

will

coincide

with

i n t h e form

N2. However,

Re(a) > >0 0.. Re(a)

if the integrand function (8.1),

we note

e(s,t)

then t h e subset A i n (8.14)

that

condition

(8.1)

is

sufficient and in certain other cases the corresponding H-functions can also be represented by the sum of several complete double series.

Example 8.2. Let H[x,y;(a,-l,l),(/3,-1,-1), (0,1,0), (0,0, l);(y,-l,l);Ls,L l);(y,-l,l);Ls,Lt]

= -^— 2

f

f T a-s+t, /3-s-t,s,tl r x-y-'dsdt.

(27ri) J J L *~s+t L

J

L

t s

64

Then above integral converges if |arg(x)| < 7i, |arg(y)| < n

I argix)-arg(y)

and

I < n.

In this case we have a+t-s, 0-t-s,s

1

rm/Sit gij 2ni

1

2ni

r(t)y

x ds y+t-s

W*

a+t+p, /3-t+p dt y+t+p

p= 0

I

(-1)PP|

a+p+t,0+p-t,t 1

y_tdt

I 2ni

p!

y+p+t

p= 0

The last integral for each p = 0,1,2,... is evaluated by two sets of the left poles t = -q and t = -a-p-q. Hence we obtain finally, that our integral is equal to

a+p-q,/3+p+q p=0

E3

)

y+p-q

q=0

a+p+q

q =0

r a

[ ;1v^™)^f+!:r]/

a+0+2p+q, -a-p-q

xr y-a-q

x H (a+/3,1+a-y, l+a;xy,-y), where |x| < 1, |y| < 1 and |xy| < 1/4, |y| < 1/2+1/2/1-4|xy| denote the following functions from Horn's list

(A.Erdelyi

V. 1) i

(8.16)

G (a,0,0';x,y) = ) 1

/

<

m

(a)

m+n

(0)

n-m

65

(0')

n

V^, ■

m-n m ! n !

; G

and H

et al. 1953,

00

(8.17)

H3(a)P>r;x,y)

-

E-

(a)

(/3) 2m+n

'—' m, n = 0

Example 8 . 3 .

m n n ^ X y .

m+n

Let

H[x,y;(a,a,A)

; ( 0 , b , B ) ;L ,L ] 1

(2rri)2J -100

1 0

0

J L

J

-100

Then this integral converges if maxj |Aarg(x)-aarg(y)|,|Barg(x)-barg(y)| V < -^- IaB-bAI.

Hence after the replacement of variables s

= as+At, t

= bs+Bt and

from Example 8.1 we obtain the following representation H[x,y; (a,a,A) ; (/3,b,B) ;L ,L J =

f

B/A x exp -x

-b/A y

x

y

-A/A a/A -x y

■ ) ■

where A = aB-bA * 0, Reioc) > 0, ReifB)

> 0.

Example 8.4. Let us

H[x,y;(a,-1,0),0,0,-1),(0,1,0),(0,0,1); (y,-l,-l),(A,l,l);L

,L 1/2

(8.18) l/2+loo l / 2 + loo

a-s,/3-t, s, t (27li) 1/2-ico 1/2-ioo

Jte(a),

y - s - t , X+s+t

Re((B)>l/2. 66

x y dsdt,

] 1/2

According to estimation (4.10) here we have

a-s,/3-t, s, t e(s,t) = r

exp

[-n(|s|+ |t|-|s+t|j]

,Re(a)-l/2

y-s-t, A+s+t Re(/3)-l/2 X

Consequently

s+t

t

from

Theorem 4.1

-Re(y)-Re(A)+l/2

we

obtain

that

the

integral

in

(8.18)

converges if and only if x,y € IR and

(8.19)

Re(a-y-A)+l < 0,

Now we present

Re(/3-y-A)+l < 0, fie(a+/3-y-A)+3/2 < 0.

H-function

(8.18) in terms of double

series. We

2

consider this question in the four domains of IR : {0<x, {0
1) The

case

0 < x < l ,

and

{Kx, Ky>.

0 < y < 1. Since Re (a), tfe(/3) > 1/2

then

integral in (8.18) can be evaluated twice from the sum of the residues of the left poles of the integrand function, i.e., at first for s = -p and then for t = -q. Finally, denoting the H-function

(8.18) by H[x,y], we

get 00

a+p,/3+q

(-l)p+q

V"

x p yq

y+p+q,A-p-q

p,q=0

a,0 F (1-A,a,0;y;x,y), where

F (a,b,b';c l

x y)=

- I

(a) p+q

(b) (b') p q

(c)

p,q = 0

P q x y

p!q!

p+q

is the first Appell series. 2). The case

0 < x < 1, l < y . Then by the replacement of t by -t in

(8.18), we get 67

-1/2+lco

l / 2 + loo a-s,/3+t, s, -t

H[x,y] = (27Ti)2

y-s+t, A+s-t

-l/2-loo

x"s(i)_tdsdt. y

l/2-loo

Hence the last double integral can be evaluated by residues of the left poles s = -p, t = -£-q as follows

ir

i

ct+p,/3+q

t-l>P*q

V

P -/3-q

x y

r

y+p-/3-q,A-p+/3+q p,q=0

a,0 y"/3Gi(a,3,l+3-y,l-^-A;xfi)l

=r y-|3,A+0 where G

l

is Horn's series (8.16).

3) The case

1 < x, 0 < y < 1. Similarly to the second case we obtain a,0

H[x,y] = r

x y-a,A+a

4) The case

a

G (a,/3,l+a-r,l-a-A;i,y). l x

1 < x, 1 < y. Then the integral in (8.18) can be evalu­

ated by residues of the right poles s = a+p, t = 0+q as follows

H[x,y]

a+p,0+q

V ("1)P*

x

-a- P -/3-q

y

y-a-0-p-q, A+a+0+p+q

p,q=0

a,/3 x %

=r A+a+0, l+a+0~r

^F (l+a+0-y,a,0;A+a+0;i,i). l x y

The values of H[x,y] for x = 1, y = 1 are defined by the continuity of the function H[x,y] and they can be obtained from Gaussian theorem (see L.J.Slater (1966)) c, c-a-b'

(8.20)

F (a,b;c;l) = V 2 1

fle(a+b-c) < 0. c-a,c-b

68

summation

§9. Characteristic of the general H-function of two variables

In this section we introduce important notion for classification of the general H-function

of

two variables. Remember

a u+A v = 0 , j

determine

in the

that any equation

j = 1,2,...,m+n,

j

(u,v)-plane

the corresponding

line L

which passes

through the origin of co-ordinates. Since any two proportionate pairs (a ,A ) 1

and

(a# ,A ) = (Aa. , AA. ) J

l

2

J

2

J

i

define

only

one

line

in

the

i

(u,v)-plane, then the exact number of these lines may be less than m+n. We shall call this number by the term of characteristic of the general H-function

of

two variables.

In accordance with

Definition

4.1

of

neighboring lines the projection to axis (Ov) of the point of intersection of line L, and upper half circle with unit radius equals to a . Hence we give the

exact definition of this notation.

Definition 9.1. If among a

in (4.14), j = 1,2,...,m+n, there exist

only r various values, then number r is called the characteristic of the general H-function (2.2) of two variables. If the characteristic of the H-function (2.2) is equal to r, then after combining all proportionate pairs for the function G(s,t) (2.3) we can always write (9.1)

G(s,t) =

r T1 T G (d s + D t), ' k=l

k

k

k

where d D -D d * 0 for k * %J j , and G (T) i s a function of one variable T of k j

k j

k

*

the type

TTr(a '_'

(9.2)

G (T) k

= - ^ n

jk

+ a T) jk

k

T T' r o jk + b jkT) 1

j=l

69

Let

r = 1,

accordance

i.e.

with

a A -A a j k

=0

j k

Corollary

for

4.4

all

the

j , k = 1,2, . . . ,m+n.

corresponding

Then

H-integral

in

(2.2)

diverges. Let now r = 2 . Then t h e H - f u n c t i o n two F o x ' s H - f u n c t i o n

(2.2) breaks into the product

i n t h e a l t e r n a t i v e form ( 2 . 1 ) a s

H|x,y; (a,a,0) L

m

,(a',0,A)

1

2

;

m

(/3,b,0)

n

of

follows

,(/3',0,B) l

n

2

; L , L.l s tJ

(9.3) = H[x;(a,a)

;(/3,b) ;L] n s

m 1

H[y; J

H f x , y ; (a, k a , qn a )

;(/3\B) mn

m

2

, ( a ,pA,hA) ^

1

(9.4)

(a',A)

1

;L.]. t 2

;

m 2

(/3,kb,qb)n , O ' ,pB,hB)n ; L Q , L J n R i 2 ° -I

= J_ lAi

H

[xh/A y-p/A; L

(«,a)

;0.b) m

; L kC+qC

i

_1 J

x H | x - q / \ k / A ; (a'.A) AP'.B) ; L +. _ 1 , L m n pc+hC J where k , h , p , q G R, A = k h - p q ^ 0 and ( a , k a , qn a ) means (a , ka , nqa ) , . . . m

(a

m

, ka

l

,qa ) , and so on. The e q u a l i t y m ^ m l

(9.4) with the l i n e a r property

Let r = 3 . Then from in the next

1

but

1

(9.2)

is

(9.1)-(9.2)

(6.4).

we can a l w a y s w r i t e t h e

form

H[x,y;

(9.5)

1

l

is evident,

l

t h e c o m b i n a t i o n of

=H

(9.3)

[w

(a,a,A)

; (0,b,B) ; L , L,. ] m n s t

(^,c,e)

m 1

(5,E,E)

n

, (a',a',0)

.(a'^O.A*)

2

;

m 3

,(/r,b',0) 1

m

2

n

70

, (/3",0,B')

n

; L' 3

,L! 1 s t J

H-function

m

m

1

m

2

3

7 7 T[y + c (s+t)] "f7 r(a'+ a's) 7 7 r(a" + A't) r

J

j=i

(27Ti)2

J

J

j=i

n

J

1

J

j=i

n

J

n 2

3

L;L' 7 7 r[5 + E (s+t)] 7 7 ro'+ b's) 7 7 r(£" + B't) x x 7y dsdt, 1

i

where x = x y , y = x y ; e, E, a* , A' , b' , B* € R; y, 6, a' , a' * , |3' , £'' € C and they are defined

from a, a, A, £, b, B ; L' , L' are new

contours. We shall study this function in detail in Chapter II. Further the G-function of two variables defined by O.I.Marichev and Vu Kim Tuan (1983) can be an example of the H-function (2.2) of two variables with the fourth characteristic. This function is represented as

follow (IK

f(

m ,n ;m ,n ;m ,n ;m ,n |(a ^ 1 J

1

2

2

3

3

4

4

pi,qi;p2,q2;p3,q3;p4.q4

(b

(1)

)

, (3). , (4).

, (2)x

),(a (b

), (a (2)

)

(b

(3

), (a

) <4

(b >)

')

x,y

(9.6) y +loo y +ioo 1 2

1 (27ii)

$ (s+t)$ (s-t)$ (s)$ (t)x "y _tdsdt, 2

1

V -ico

2

3

4

r2-- 1 0 0

where b

(9.7)

* k

(T)

1

+T, .

= r b

n +i k

+T,

v< k

b m

^

(k)

k

,a ( k ) + x, P

-

+x, 1-al -x,
* <

1-b

k

k

>

m +i k

-T,

-

k

, 1-a n

-T

k

,l-b ( k , -T

1,2,3,4. Finally, the H-function (1.14), defined by P.K.Mittal and K.C.Gupta (1972) is an example of H-function (2.2) with a characteristic r £ 4.

71

CHAPTER II. THE H-FUNCTION OF TWO VARIABLES WITH THE THIRD CHARACTERISTIC

§10. Definition and Notation In Chapter I, §9 we have

introduced the notion of the character­

istic of the general H-function of two variables and showed that this function is of type (9.5) when its characteristic is equal to three. In this chapter we use the following notation of the H-function with the third characteristic.

Definition 10.1. H-function H(x,y) of two variables with the third characteristic is called the next function (1)

(

m ,n :m ,n ;m ,n 1 1 2 2 3 3 x,y P 1 ,q i :p 2 ,q 2 ;p 3 ,q 3

(a

UK

,a

(2)

f

): (a

(2K

,a

,

(3K>

(3)

); (a

,a

)]

(/3(1),b(1)):(/3(2),b(2));0(3),b(3)) q

q

i

i

q

2

q

2

q

3

q

3

(10.1)

$ (s+t)$ (s)$ (t)x ^ ^ d s d t , 1

{2niV

2

3

L L 2

1

where

(10.2)

$ (s+t) =

m n rtr(p < 1 ) + b < 1 ) (s + t)]TTr(i-a < 1 , -a ( 1 , (s + t)| J j j=l Ij Jj = l I J '

Y\ j=n +i

rfa!1,+a;i,(s+t)] TTrfi-P^-b^ts.t)) ^

J

j

72

' j=m +i l>

'

m

n

urfrw?.) rp-KS"') 2 /

(10.3)

$ (s) 2

\

p

2

q

-TTrf«(2>+a(2)s) 7 7 rfi-p (2, -b (2, sl

m

n

rpK*."'] 17H"-i"«) 3

(10.4)

$ (t) 3

/

3

N

P,

q3

rrr( a ; 3, + a; 3 't)rTr(a- P ; 3> -b; 3 't)

j=n + i v

J

J

J

> j=m + i ^

J

'

where x and y are not equal zero and an empty product is interpreted as unity. Also, the non-negative integers m that O ^ m

k

^ p

Greek letters a

(j = 1,2

k

(k=l,2,3) are such (k)

,b

are all complex numbers.

is in the s-plane and runs from f -ioo to f +ioo and poles

of

(j = l,2,...,n2)

T 1-a

from

the

-a

(s+t)

poles

(j = l,2,...,n ),

rf/3(1)+b(1) (s+t)l

of

m ), riy 2) +b| 2) s] (j = 1,2,...,m ), when t € L . *

^ J

The contour L

J

J

2

2

is in the t-plane and runs from f -ioo to f +ioo and 2

separates

are all Fpositive,

(k )

, £

the

r(l-a<2)-a'2)s]

,q

^ q ; Latin letters a

k

The contour L separates

,p (k)

,0 ^ n

k (k )

,n

the

rflV 3 ) -a^ 3 ) tl

2

poles

r(l-a (1 *-a (1} (s+t)1

of

(j = l,2,...,n3)

(j = l,2,...,m ), r(/3(3)+b(3)tl *

Y,

■*

■*

from

the

(j = 1,2,

poles

2

(j

=

l,2,...,n),

of

T^ ( 1 ) +b ( 1 ) (s+t)l

m ) when s € L .

J

Remark 10.1. If we take m

=n

=p

=q

=0

in (10.1) then

H(x,y) breaks into the product of two Fox's H-functions (1.5)

73

, 2

2

3

2 *3

3

, 2

p

2 n

*2

(/3(2>,b<2,); (P

,a

(0

2

2 (2

p

\b

3 n

*3

p

(0

3

3 ,3)

(3)

3

P >q

)

)b

(a

m ,n

)

p

3 (3)

,

)

2 (2)

,a

P 2

(2) ( 2 ) .

(a

, (3) ( 3 ) .

); (a

p 2

3

m ,n

(2).

,a

p

0,0 :p ,q ;p n,q x,y n *2

(2)

(a

0,0 :m ,n ;m ,n

)

<3K

,a p

3 (3)

,b

)

3 (3)

)

Also, if in the integral (10.1) were place s+t by s and t by t then we obtain the following formula in the case

m = n = p = q = 0

&

2

f 1 1

3

3

i

**3

(|3

3

(1)

l

r (a

l

P.

P >q *i

,b q

P

(P

):

3

(1),

,

b

3

(3)

(a

3

P >q n

*3

( 3 ) q

i

m ,n

P1

P

3

(3)

,

n ,a )|

2

i (1)

i

M

*2

(a(3), a<3)))

(1

C0(1).b

l

p

i (1)

q

m ,n

(IK

,a ):

p

x,y

p ,q :0,0;p ,M q M *i

(1)

(a

m ,n :0,0;m , n

2

(P

3

>

(3),.,

,a P

3 ,3)

)

3

P

) 3

,b(3))

Similarly, we have ,

i

*2

, (2)

): (a

l

(1) ,(1)

2

): (P

<2>

= H P >q

(/3U,,b

2

(1),

,b

f

(2),

(2)

(a

m ,n

1

(2K

,a );

l

(1) r(a(1) ,a n )|

m ,n l

O

2

(IK

,a l

1 1 2 2 p ,M q :p M ,q ;0,0x,y *i

(1)

(a

m , n :m ,n ;0,0

(2) . A

,a

)1 2

2

P »CI

(|3(2,,b

(2)

*2 ^2

Remark 10.2. In the case, when m = 0 (or n = 0 ) , the H-function l

(10.1) coincides with R.S.Pathak S.L.Kalla

the functions

(1970), K.K.Chaturvedi (1971),

S.L.Bora

l

of two variables

and A.N.Goyal

and S.L.Kalla

(1970),

R.U.Verma (1971), M.Shah (1973a,b) (see also (1.11)). 74

introduced by

(1972), P.C.Munot and R.K.Saxena

(1971),

§11. Convergence Theorems

Now f o r d = f + f 1

(11.1)

, d = f , d = f

1

2

K = -^-

2

1

3

2

k k ^ , (k)^ „

(k)

m

n

>b

+ Fa

J

and k = 1 , 2 , 3 d e n o t i n g &

p „

-

q

k

y

a

J

j=i

(k)

„

-

_

/i = Re

j=i

^k

(k)

j=i

„

0

j

j=n +i k

(k)

, (k)

b

J j=m +i k

P (11.2)

k

Y

+ d

j=i

9 k

„

(k) j

U

J=l

p - q

k ~ ^(k) j J=l

*k

^k

and making use of the general asymptotic estimation (4.10) obtained in §4 we have

|* (s+t)$ (s)$ (t)x sy"t

(11.3.)

« S exp

|U+V|+K

[- ■(-.

"M2

"**1

x |u+v+i |

|U|+K |v|| + uarg(x)+varg(y)

|u+i|

^3

|v +i| 2

2

where H = const, s e L , t e L , u = Zm(s), v = J/n(t), u + v 1

2

from Theorem 3.2 and Remark 3.3

Theorem 11.1. The integral

—>oo. Hence

we obtain the following results.

in

(10.1) converges

if,

f |arg(x) | < 7r(/c + K ) , 1 1 2

(11.4)

Urg(y) | < 7T(K + K ) , |arg(x)-arg(y)| < TT(K + K ) .

7f we replace one

(>) for at least

the symbol one above

(<) in conditions inequality,

diverges.

75

then

(11.4) by the the

integral

opposite

(10.1)

will

Consequence

11.1

If

K + K > 0,

n

integral

1

(10.1) converges

for

all

K + K > 0,

2

1

real

K + K > 0, 2

3

positive

variables

then

the

3

x and y.

Further, here we establish the following auxiliary lemmas for the wording of more strong criterion of the convergence of the

integral

(10.1) in the case x,y e (R+ .

Lema 11.1. Let

B,C,a,b,c 6 R and i = "/-T . Then

lu+v+i | a |u+i | |v+i|° exp(Bu+Cv)dudv < oo

(11.5)

if

all

and only

if

f 2sgn(B) + sgn(a+b+l) < 0 , (11.6)

2sgn(C) + sgn(a+c+l) < 0 , 2sgn(B) + 2sgn(C) + sgn(a+b+c+2) < 0

Proof. It is showed by Lemma 4.1, Chapter I, §4, that the integral (11.5) converges

if and only

if one of

the following

four

conditions is true:

1)

B < 0, C < 0.

2)

B < 0, C = 0, a+c+1 < 0.

3)

B = 0, C < 0, a+b+1 < 0.

4)

B = C = 0, a+b+1 < 0, a+c+1 < 0, a+b+c+2 < 0.

Evidently it is equivalent to the inequalities (11.6).

Lemma 11.2. Let all (11.7)

A,B,C,a,b,c € R , i = V^l and

Q(u,v) = |u+v+i|a|u+i|b|v+i|° exp(A|u+v|+B|u|+C|v|).

Then

76

sets of

(11.8)

fi(u,v) dudv < co

<=»

ur f 2sgn(A+B) + sgn(a+b+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 , 2sgn(C+A) + sgn(c+a+l) < 0 , 2max jsgn(A+B)+sgn(B+C),sgn(B+C)+sgn(C+A),sgn(C+A)+sgn(A+B)t + + sgn(a+b+c+2) < 0. Proof. As is known from Chapter I, §4, three

lines represented by

the equations u+v = 0, u = 0, v = 0, separate the (u,v)-plane into six nonintersecting sectors W ,...,W . In the sector °

1

6

W = { ( u , v ) € IR , u > 0, v > 0} from Lemma 1 1 . 1 we h a v e Q ( u , v ) dudv

lu+v+i | a |u+i | |v+i|c exp(A|u+v|+B|u|+C|v| ) dudv < co o o if and only if ( 2sgn(A+B) + sgn(a+b+l) < 0 , I 2sgn(A+C) + sgn(a+c+l) < 0 ,

(11.9)

[ 2sgn(A+B) + 2sgn(A+C) + sgn(a+b+c+2) < 0 Further, as in §4, if we replace u+v by v sector W

and -u by u

then

= {u,v € IR , u < 0, v > 0, u+v > 0} breaks into W . Hence 2

Q ( u , v ) dudv

1

Q(u +v , - v ) du dv i

l

l

1 1

77

the

| u +v + i | b | u + i | a | v - i | c e x p [ ( A + B ) |u | + (B+C)|v 1

1

1

1

1

I )du dv i

l

l

< oo

O 0

if and only if

f 2sgn(A+B) + sgn(a+b+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 ,

(11.10)

2sgn(A+B) + 2sgn(B+C) + sgn(a+b+c+2) < 0

Similarly,

fi(u,v) dudv < oo

(11. 11) W

2sgn(C+A) + sgn(c+a+l) < 0 , 2sgn(B+C) + sgn(b+c+l) < 0 , [ 2sgn(C+A) + 2sgn(B+C) + sgn(a+b+c+2) < 0 Note that the double integrals from the Q(u,v) on the domains W , 4

W , W 5

converge provided conditions (11.9), 6

respectively.

Finally,

(11.10),

(11.11) hold true

the set of these conditions

is equivalent

conditions (11.8). This completes the proof of Lemma 11.2. Lemma 11.3. Let, as in Lemma 11.1, all

(11.12)

sup

i lu+v+i |a|u+i |b|v+i |cexp(Bu+Cv)i < oo

J

f 2sgn(B) + sgn(a+b) ^ 0 , 2sgn(C) + sgn(a+c) ^ 0 , 2sgn(B) + 2sgn(C) + sgn(a+b+c) < 0 Proof. At first, note that the function

(11.14)

Q (u,v) = |u+v+i| |u+i| |v+i| exp(Bu+Cv)

78

■

B,C,a,b,c € !R and i = -/-F. Then

u,v > 0 v

(11.13)

to

is continuous in the plane R . Hence the inequality (11.12) holds valid if and only if

(11.15)

lim Q (u,v) < oo o

2

for u +v

2

—> oo .

Further, it is not difficult to obtain the following statements:

i) Let B = 0, C < 0. Then the inequality (11.15) is true if and only if

a+b < 0.

ii) Let B = C = 0. Then lim Q (u,v) < oo o

2

for u £ v, u +v

2

—> oo ,

if and only if a+b ^ 0 and a+b+c ^ 0.

Now from the above statements i) and ii) we can prove the inequality (11.12) holds valid if and only if one of the following four sets of conditions is true 1)

B < 0, C < 0.

2)

B < 0, C = 0, a+c ^ 0.

3)

B = 0, C < 0, a+b ^ 0.

4)

B = C = 0, a+b < 0, a+c < 0, a+b+c ^ 0.

But it is equivalent to the conditions (11.13).

Lemma 11.4. Let

(11.16)

Q(u,v) be defined

sup

by (11.7).

Q(u,v) < +oo

Then

«=»

U, V > 0

2sgn(A+B) + sgn(a+b) ^ 0 , (11.17)

2sgn(B+C) + sgn(b+c) ^ 0 , 2sgn(C+A) + sgn(c+a) ^ 0 , sgn(A+B) + sgn(B+C) + sgn(C+A) + 2sgn(a+b+c) ^ 0.

79

This Lemma can be proved by analogy with the proof of Lemma 11.2. Here it is sufficient to note that if A+B ^ 0, B+C ^ 0, C+A ^ 0, then the condition

(11.18)

2 maxisgn(A+B) + sgn(B+C), sgn(B+C) + sgn(C+A),

sgn(C+A) + sgn(A+B)l + sgn(a+b+c) ^ 0

is equivalent to

sgn(A+B) + sgn(B+C) + sgn(C+A) + 2sgn(a+b+c) ± 0.

Now from the asymptotic estimation (11.3) and Lemma 11.2 we obtain the following theorem about the convergence of the integral in (10.1).

Theorem 11.2. The integral variables

x and y if and only [

(11.19)

in

(10.1) converges

for

real

positive

if

2sgn(K + K ) + sgn(ji + jn.-l) > 0, for k,j = 1,2,3 and k * j;

2min «|sgn(K + K )+sgn(K + K ), sgn(/c + K )+sgn(»c + K ), ^ 1 2 2 3 2 3 3 1 sgn(/c + K ) + sgn(K + K ) \ + sgn(a + u. + a - 2) > 0. 3

1

1

2 J

1

2

3

We conclude by remarking that Theorems 11.1 and 11.2 immediately follow from Theorems 3.1 and 4.1 respectively, as particular cases.

§12. Reduction formulas for the H-function with the third characteristic

The H-function with the third characteristic (10.1) as a particular case, possesses all properties in § 5 — § 8 for

the general H-function

(2.2). In this section we consider the cases when the H-function

80

(10.1)

of two variables reduces to Fox's H-function These

properties

are usual

only

(1..5 of one variable.

for the H-function

with

the third

characteristic. Here we assume additionally integral

that

in (10.1) are the vertical

the contours L lines with

and L

i

in the

2

real parts c and C

resperalvely, i.e.

L

l

s LL

, c = {|1sseeC C,

Re(s=cc

1, J1,

L L2 =L Lcc = =j jt t€ € C, C, Rei)) Rei)) = =C C 1. 1. 2 =

First we establish auxiliary result.

Lemma 12.1. Let c > 0, C > 0. Then

(12.1) (12.1)

(27Ti)

(12 2) (12.2)

J J L C Lc

1

1

formulas

are

valid

0(x)r(T)(x+y)~TTdr, >(s+t)r(s)T(t)x"V t dsdt = -!— 0(x)r(T)(x+yr

— 1 — —1— 2

the following

2711 J L C+c

[*Cs+t)x]__[f 00((TT)) tmax(xty)]^ tmax(xty)]^ ff [*te+Uxy-tdsdt == __J y -tdsdt tt TT

(27ii )' JJ (2TTI)' L

27ri JJ 27ri

Js s J

L

C

c

"C+c

where T(s) is

the gamma-fund gamma-fundtion, ion,

the ineegrals

are convergent

{in

0(T) is some function (12.2) it

such

is assumed that

that

all

of

x,y > 0).

The proofs follow from the change of variables s by s , s+t by t , after which the following integrals are used r(p-s)T(s)z"sds = r(p)(l+z)fpP,

—— ? —— 2711

L

1

c ('

s f _ J I z"z" s !S

p ~\ ,

—T T " TFiTids (p-s)s ds " I 2711 2711

0 < < c c < < ReRpp; ReRpp;

J

II

c c

if z < 1,

P [[ P"V .. P _1 zP p ,, if if z z> > . . 81

0 < c < e(Reip). 0< c<* p).

The last two equalities are easily obtained with the help of the residue theory (see, for example, O.I.Marichev

(1983)).

From (12.1) it follows immediately the following reduction formula of the H-function (10.1) to the Fox's H-function (1.5) of one variable (a ,a ):

m,n:1,0;1,0

p

p,q:0,l;0,l x,y

P

(0 ,b ): (0,1);(0,1) q

q

(12.3) m+1 ,n

= H!p,q+i

(a ,a ) p P

x+y

(0,1), (0 ,b )J q

Here for

q

y

simpleness we replace m ,n ,p ,q

by m,n,p,q,

respectively.

Similarly, as a consequence of the equality s = r(s+l)/r(s) it follows from (12.2) that (a ,a ): (1,1);(1,1))

m,n:1,0;1,0

P

p,q:l,l;l,l x,y

P

(0 ,b ): (0,l);(0,i: q

q

(12.4) (a ,a ),(1,1) m+l,n p P max(x,y) p+l,q+l (0,1), (0 ,b )J q

(x,y > 0 ) .

q '

Naturally we will call (12.3), (12.4) respectively the sum and maximum properties for the H-function (10.1) with the third characteristic. Now we consider reduction cases for this H-function when y = x. Lemma 12.2. Let

equality

(12.5)

is

f(x)g(x) = h(x) for

all

x > 0. Then

true 1

^(s+t)f*(s)f*(t)x"s_tdsdt

(27ii r L

CLc {r)h (T)X dx,

2ni "C+c 82

the

following

where f ,g ,h

are

respectively

the Mel 1 in

transforms

of

the

functions

that

all

integrals

f,g,h

(12.6)

f (s)

Moreover,

the

functions

are

f(x)x

assumed

dx.

to

be such

converge. The

proof

of

(12.5)

is obtained

variables T = s+t along with

by again

the complex

using

the change of

convolution

for the Mellin

transforms 1 2ni

h (z)

Example (1+x)

12.1. If, in Lemma

, then h(x) = (1+x)

f (z-t)g (t)at.

12.2, we set f(x) = (1+x) , g(x) =

and consequently

Rein)

f*(s) = r(fi-s)r(s)/r(jz), with

* * g and h similarly

expressed.

Hence

> o

we obtain

the reduction

formula

p

p,q:l,l;l,l

O

p

'

,b ): (0,1);(0,1) q

T{u)r{v) r(u+v)

tt-vtD)

(a ,a ): (l-ji.l);

m,n:1,1;1, 1

q

m+l,n+l "* p+l,q+l

w

(a ,a ) , { l - u - v , l ) } p

p

'

(o.i), o q

Example h(x) = (l-x)^

12.2. Let +y 2

Reiii),

g(x) = (1-x)^

, where (l-x) a , if x < 1, (1-xr = , if x > 1.

Then, since

f (s) = r(fi)r(s)/r(M+s), 83

> 0.

q

f(x) = ( l - x ) M 1 ,

0

Re{v)

,b )

Rein)

> o

1

and

then

from formula (12.5) we have (a ,a ): (jn. 1); iv, 1))

m,n:1,0;1,0

p

H p,q:l,l;l,l

p

'

(0 ,b ): (0,1);(0,1) q

q

(a ,a ), (iLt+i^-1,1 )^

r(jLL+i^-l) j, m+l,n r(ji)r(v) H p+i,q+i

p

p

(0,1), o ,b ) q

Example

Re(fi), Re(v),

'

Re{n+v-l)

> 0.

q

12.3. For generalized hypergeometrie functions

(1.2) the

following formula is well-known (see A.Erdelyi et al. (1953)) F(M;x) F ( V 5 X ) =

(12.7)

F (M2,^-l);

4xl

where A(k,a) abbreviates the array of k parameters a a+1 a+k-1 k' k ' ' " ' * k Further for convenience we use the notation (a)

T(a )...T(a )

(0)

r(s )...ro )

1

0,...,0 1

i

q

Then for the hypergeometric function

p__

F

q

(a) ; (b)

;-z

we have the

P p+i | _ p p+i J following representation (see O.I.Marichev (1983), A.P.Prudnikov et al

(1989))

(12.8)

F P

[(a) ;(b) ;-zl p+i |_ p p+i j

r f ba ,...,b p+1

[ aa L

a i

(12.9)

x

s-i „ F P P+1

1

1

I

f a -s,..., aa -s,s -s,s ] -Sj p |z ds, -s, . . . , b b- -s s J

J -251 J1 *b J

L

p+i

P

[r\ t^\ 1j _|" b , . . . ,b ,a -s, . . . ,a -s,sl 1(a) ;(b) ;-x dx = T l' p + i' l P a , . ,,a ,b -s,...,b -s l P P +1 i_P + I p i_i L J L a a.b-s b -s J 84

Consequently,

from

(12.5),

(12.7),

(a ,a ): p P

m,n:1,0;1,0 (12.10)

H

p,q:0,2;0,2

(12.9)

get

—

;

—

(|3

q

,b

q

):

(0,1-/LI);(0,1-I>)

(-JLI-I>,2), (a ,a )

m+l,n+l ■• p + l , q + 4

p

p

(0,1), (0 ,b ),(l-fi,l), (1-1^,1), (-II-I>,1) q

q

is shown above, any equality of type

corresponding (10.1) with

reduction formula of the

third

hypergeometric series

F

(12.7) shall determine

(12.10) for

the H-function

particular, for Gaussian

we have the following relations

V-A 1

type

characteristic. In

' A,n;

F

(12.11)

)

x,x

u

As

we

X

V-A,i>-/i;

2 1

0

F

X

2 l

X

v

;

>

r

A, H (12.12)

F

X

2 1 [A+ii+1/2

2 1

A+/I+1 / 2

A.fl

2 1

2 1 A+ii-

A+tx+-^;

2 1 A+LI- — ^ 2

; *

J '

2 1 A+fx-

3

2

( 2A,2ii-l,A+fi-l 3 2[A+ii-—

'

2 1

- Xf *

; J

—-A-M

1

F 3 2

^

2

^

2

2

A+LI+ — , — - A - i i ^ 2 2 ^

85

;

I

J

' X

2 A+J1+— , 2 A + 2 i i - l ;

;

(12.15) =

;

F

.

— -A — 2 ' 2

F

2 1 A+ii+ -

2A,2ii,A+fi =

;

J

;

F

A,M ;

f F

X 2 A4 i i + l / 2 , 2 A + 2 / j i;

' X

F

X

F

3

;

A,ii-1 (12.14)

F

;

F

F

=

X

•

A,M ; (12.13)

2A,2ii,A+ii

i ]

,2A+2JLI-2;

}

[A-— ,fi-— ;

1

2

F

F 2 1

^

2

X

1

2 1

A+jLi- -

'

^ 2

(12.16) f

2A,2/i,A+n

;

F 3 2

A+u+— ,2A+2fx-l;

(1 A + — ,ii- — 2 ^ 2

F 2 1

J

(A+ — , u + — ;

;

2

F 2 1

A+/1+

^

2

X+V+—

; J

(12.17) f 2A+1,2/I,A+M

;

F 3 2

(A.A+

(12.18)

2 1

A+n+ — ,2A+2fi;

— 2

F,

} X

• X

2 1

2A+1

>

V.M+-JF

(A+/1, A+/1+ — ; ] x 1 [ 2A+2fi+— ;

= F

X

2

2/1+1

Here (12.11) is a well-known formula obtained by twice applying the Boltz's formula

f a,b ; ) z

F

f a,c-b ; = (l-z)' a

z-1

2 1

2 1

and

F

(12.12)-(12.18) are Slater's theorems

(see L.J.Slater

(1966) and

also H.M.Srivastava and P.W.Karlson (1985), formulas 1.3.34-1.3.42).

Now from (12.5)) and (12.11) - (12.18) we obtain

the corresponding

reduction formulas for the H-function (10.1) of two variables

(a ,a ): (l-v+jbi+X,l); (l-v+A,l), (l-v+jbi,l)l

m,n:l,l;l,2 (12.19)

H

P

p,q:l,l;2,2

(0 ,b ) q

A,fi, y-fi-A

=r

H

v-A,

v-\±

p

'

(0,1)

;(0,1),(1-v.l)

q

m+l,n+2 p+2,q+2

(l-i^+A,l), (1-V+JLI,1), (a ,a )] p p '

(0,1), (0 ,b ), (1-1^,1) q

86

q

(a ,a ):

m,n:l,2;l,2 p,q:2,2;2,2

( 1 - A , 1) , ( 1 - j i , 1)

;

( 1 - A , 1) , ( 1 - j i , 1)

(0 ,b ): ( 0 , 1 ) , ( — - A - M , 1 ) ; ( 0 , 1 ) , ( — q

q

2

)

-A-jx,l)

2

A,M,A,M,2A+2/LI

(12.20) 2A , 2 p i , A+/J, A + f i + — j

X

( 1 - 2 A , 1 ) , (1-2/1,1), ( l - A - ^ , 1 ) , (a

m+l,n+3 p+3,q+3

u H

(0,1),(/3

,b q

(a ,a ):

m, n : 1 , 2 ; 1 , 2 p,q:2,2;2,2

,a p

)

1

P

), ( — -A-^1,1), (1-2A-2JLI,1) q

2

( 1 - A , 1 ) , (1-jix, 1 )

;

( 1 - A , 1 ) , (1-JLX , 1 )

)

x,x (/3 , b ) : ( 0 , 1 ) , ( 3 / 2 - A - / L t , l ) ; ( 0 , 1 ) , ( — q

q

2

-A-^,1)

A,fi,A,fi,2A+2/i-l (12.21)

=r

2A,2/j,A+fi,A+^-

X

u H

—

J

(1-2A,1), (1-2/1,1), ( l - A - / i , l ) , (a

m+l,n+3 p+3,q+3

( 0 , 1 ) , (/3 , b a

(a

m,n:1,2;1,2

,a

):

p

p,q:2,2;2,2

,a p

(1-A,

)

)

p

), ( — -A-/1.1), (2-2A-2/i,l) a

1),

2

( 1-JLI, 1)

'

;

(1-A,

1) , (2-JLI, 1)

} '

p

((3 , b ) : ( 0 , 1 ) , ( 3 / 2 - A - f i , l ) ; ( 0 , 1 ) , ( 3 / 2 - A - J L I , 1 ) q

q

A,/I,A,/I-1,2A+2/LI-2

(12.22) 2A,2/i-l,A+/i-l,A+/i--

( 1 - 2 A , 1 ) , (2-2jLt,l), ( 2 - A - f i , l ) , (a

m+1,n+3

p,q:2,2;2,2

(0,1), O

(a ,a

m,n:1,2;1,2

,a p

x H p+3,q+3

P

):

q

,b

q

)

), ( — -A-fx,l), (3-2A-2*i,l) 2

( 1 - A , 1 ) , ( 1-JLI, 1 )

P

) p

; ( — +A, 1 ) , ( — +|Lif 1 ) 2

2

^

x,x (0 , b ) : ( 0 , 1 ) , ( — - A - M , 1 ) ; ( 0 , 1 ) , ( - — +A+/i,l)

87

-A.--M,

A , jLl,

(12.23) A-JLI+ — , f i ^

(_L-A+ii,l)f ( J--M+X,l),( — , l ) , ( a

m+1,n+3 x H p+3,q+3

2

2

2

,a P

)) P

( 0 , 1 ) , ( 0 , b ) , ( — - A - | U , 1 ) , ( - — +A+JU.1) q

q

2

2

(a , a ) : ( ^ - - A , l ) , ( — - f i , l ) ; ( — - A , l ) , ( — - ^ 1 , 1 ) ) p p 2 2 ^ 2 2 '

m,n:l,2;l,2 p,q:2,2;2,2

(0 , b ) : ( 0 , 1 ) , ( — - A - f i , l ) ; ( 0 , l ) , ( — - A - / i , l ) q

q

2

2

A - — , n - — ,A+l/,*i+ — ,2A+2ji-l' (12.24) 2A,2fi,A+ji,A+jLi- —

m+l,n+3 x H p+3,q+3

(1-2A,1), (l-2jn,l), ( 1 - A - M , 1 ) , (a ,a ) ) p p ( 0 , 1 ) , (0 , b ) , ( — -A-/U, 1 ) , ( 2 - 2 A - 2 M , 1 ) J q

q

2

(a , a ) : ( J - - A , l ) , ( — - J I , 1 ) ; ( — - A , l ) , ( — - , 1 , 1 ) ] p p 2 2 2 2 '

m,n:1,2;1,2 p , q : 2 , 2 ; 2 , 2 x,x

(0 , b ) : ( 0 , 1 ) , ( — - A - f i , l ) ; ( 0 , 1 ) , ( — -A-,1,1) q

A+—

,H~—

q

2

2

, A + 1 / , M + — ,2A+2/i"

(12.25) 2A+l,2jn,A+fi,A+^+ ■

m+l,n+3 p+3,q+3

(-2A,l),(l-2/i,l),(l-A-/i,l),(a p

( 0 , 1 ) , (0 , b ) , ( — - A - j y i , l ) , ( l - 2 A - 2 f i , l ) q

m,n:l,2;l,2 p , q : 2 , 2 ; 2 , 2 x,x

,a ) p

q

2

(a , a ) : ( 1 - A , 1 ) , ( — -A, 1 ) ; (l-^i, 1 ) , ( — - j i . l r t p p 2 2 ^ ' (0 , b ) : q q

( 0 , 1 ) , (-271,1)

88

;

( 0 , 1 ) , (-2ji, 1)

A , J L I , A + — ,li+— (12.26)

=r

,2A+2|i+ —

\+li, A+fi+ — 2A+1, 2/n+l

(1-A-/LX, 1), ( —-A-[z,l),(a ,a ))

m+l,n+2 x H p+2,q+2

2

P

P

(0,1), (0 ,b ),( 4--2A-2fi,l) I q

q

2

'

§13. The G-function of two variables and its special cases

In this paragraph we introduce the G-function of two variables, which is a special case of the H-f unction (10.1) of two variables, when all coefficients

a

, b

, k

1,2,3,

are equal to 1

,

(13.1)

1

1

2

2

3

3

s

M

l

*2

M

2 *3

M

):

(0(1)):

3

m ,n :m ,n ;m ,n

H

1 1

2

2

3

3

*1

l

M

*2

2

1

(2),

*V

M

0(2));

(a (|3

(3K 3

^

) }

(3),

(a(1),l):

(a(2),l);

(a(3),l) 1

3

(/3(1),1):

(0(2),1);

(0(3),1)

* (s+t)¥ (s)¥ (t) x sy

(27Ti) 2 , I-

.

);

x,y

p ,q :p ,q ;p ,q M

,

(a

x,y

p ,q :p ,q ;p ,q *1

(1).

(a

m , n : m , n ;m , n

1

2

\isdt,

3

I

2

"l

where

77 (13.2)

* (x) =

ro! k ) + x)

^r(i-ajk)-x

- ^

j= l

Pk

k

k

77 j=n

+i k

r(a (k) +x) J

k =

7 7 rd-/3 ( k ) -T) j=m +i k

89

J

1,2,3.

Here, as in Definition 10.1, for the H-function with the third characteristic, the contour L runs from f -i« -i« to f +ioo, +i«, the contour L runs € • * rf2 +i■ • tl i ' HH t-planes t i tf fa-i' to t f ' in ' the th complexs1 1 * ' respectively. f i 1 i-t ft , from f and Likewise from « , to f2+i~ m the complexsand t-planes Likewise theseV icontours satisfy^ the known condition of respectively. poles separation with these contours satisfy^ the known condition of poles separation with In particular, if i^ = 0 (or i^ = 0) our G-function (13.1) coincides with the Agarwal's G-function (1965) and Sharma's S-function (1965) (see also (1.8)). The

convergence

theorems

for the G-function

(13.1)

immediately

follows from Theorems 11.1 and 11.2. Here in accordance with (11.1), (11.2) we denote (13.3) (13.3)

cc = = m m + + n n -k k k k k k

pk

Pk+q p +qfc —■=— , V^ . 2 2

qk

(k) (k) ((kk)) y y = Re R e V> a ct -- V) 0 0 + (d — ) )(p -q ), ), kk == 1,2,3, 1,2,3, j j k 2— k (p k -q

(13.4)

kk

\ L^ L*

^ j=l j=i

L^

j=l j=i

J'

where d = f +f , d = f and d = f . Consequently, according to (11.3) 1

1

2

2

1

we obtain the estimation

3

2

n

I* (s+t)¥ (s)¥ (t)x"sy't| - S 2 exp \-nlc 1

1

2

3

LI L I1 1 -r

J

* ' 1

+ uarg(x) + varg(y) |u+v+i|

-r1x -y

| \i+i |

J

»

o

*

|u+v|+c |u|+ c Ivl] 22

33

J

-y -y |v+i|

+ uarg(x) + varg(y) |u+v+i| | \i+i | |v+i| for u = Jm(s), v = Izn(t) as Izn(s), J/n(t) —> <x>,s € L , t € L . 1

2

1

2

for u = Im(s), v = Jm(t) as Izn(s), J/n(t) —> oo,s4- V»*a € /"" L f,n nt n +€ i L . Mr\t F ^ FA rr l \ / £ i +•}■»£» r i r \ r r o c n r \ r » r l i n n 4" V» A A T - « m e * -f n r An Mr\t F \ FA

rr l \ / £ i

+•}■»£»

rir\rrocnr\r»rl i n n

Theorem 13.1. The integral

4" V» A A T - « m e *

- P A T - 4- V»*a C

in (13.1) converges

f n n n + i An

if, if,

f |arg(x)| < 7i( c 22)) , , 7i(CCii + c (13.5)

|| |arg(y)| < rct^ 7r( C i + C c33) , (( |arg(x) |arg(x) -- arg(y)| arg(y)| << TT(C TT(C22 ++ cc33)) . . 90

f 11

i \

f 11

i \

If

we replace

one (>) for at least

the symbol

(<) in conditions

one above inequality,

(13.5) by the

then the integral

opposite

(13.5) will

diverges.

Consequence 13.1. If n

integral

(13.1) converges

c +c

> 0, c +c

1 2

1 3

for all

Theorem 13.2. The integral variables

x and yif \

(13.6)

and only

> 0, c +c 2

real positive in

> 0, then

variables

(13.1) converges

the

3

x and y. for

real

positive

if

2sgn(c +c . ) + sgn(r +y -1) > 0,

for

k,j =1,2,3 and k * j;

d i n <-|sgn(c + c )+sgn(c + c ), sgn(c +c ) + sgn(c +c ), 2min

sgn(c + c ) + sgn(c + c ) > + sgn(^r + y + ^ - 2) > 0 .

Now we assume that the contours L L

c

= {Reis)

= c> and L = L^ = {Reit) 2

and L 1

are vertical lines L = 2

l

= C}. Then, any reduction formulas

C

in §12 for the H-function (10.1) will similarly define the relations for the G-function

(13.1). For example, from

(12.3),

(12.4) we obtain the

sum and maximum properties for the G-function (13.1) of two variables as follows:

m,n:1,0;1,0 (13.7)

p , q : 0 , l ; 0 , l x,y

(a ): —

; —

p

(0 ):

0

0

^m+1, n ' p >q + i x+y

q

m,n:1,0;1,0 p,q:l,l;l,l x,y

(a )

0, o ) q

(a ): 1 ; 1 ) p

(0 ): 0 ; 0 q

(13.8) (a ), 1 m+1 ,n = GJ max(x,y) p+l,q+l

(x,y > 0 ) .

0, (0 )J q

91

y

Since for Y.L.Luke

the Meijer G-function (1969), O.I.Marichev

there are many particular

cases

in

(1983), A.P.Prudnikov et al.(1989), then

the two last reduction formulas give the corresponding particular cases for the G-function (13.1) of two variables.

For instance, from

_1,Of I - 1

G

x

o,i[ | 0 J

-x

=e

•

0,0:1,0;1,0 we obtain

0,0:0,1;0,1 x,y

—

: 0 ; 0

and

-

1,0:1,0;1,0

:1 ;1 1 = G 2'°(max(x,y)| o ^

0, 1: 1, 1; 1,x,y 1

1

]

: 0 ; 0

= exp[-max(x,y)]. The last equality can be written as follows r(i+s+t) -s -t , ,. x y dsdt =

1

{ZniV L

where Re is)

, ,, r exp[-max(x,y) ] ,

CLc

= c > 0, Re{t)

= C > 0.

By applying Theorem 7.3 on the double Mel1 in transform from the general H-function we obtain 00 00

3Tt Jexp[-max(x,y) ] ; s,t

r(l+s+t) st

exp[-max(x,y)] x

Re{s)t

Reit)

y

dsdt

> 0.

Consequently, we get the accordance of the double Mel1 in transform and its inversion 92

> r ( 1 ^ + t ) ,,

exp[-max(x,y)] <

Reis)

Rei))

> 0,

> 0.

For convenience here we construct the table of particular cases of the G-function (13.1) of two variables of the last relation type. Further in all the next formulas Reis) the

integral

exponential

Viv,x) bility's integral; Viv yx)

function;

Re i)) > 0, Rei))

erfc(x)

> 0 and Ei(x) is >

is the additional

proba-

is the non-complete gamma-function; J (x) is

¥(a,b;x) is the the Bessel function; K (x) is the Macdonald function; *(a,b;x) degenerate

Tricomi

function

(see

A.Erdelyi

et

al.

(1953)

and

A.P.Prudnikov et al. (1989)).

TABLE 13.1

h*(s,t) = |h(x,y)x h(x,y)xs-1yt"1dxdy

h(x,y)

II

e" x " y

rfs.tl, rls.tl, Re(s) > 0, Re(t) > 00

2

exp[-max(x,y)l exp["-max(x,y)l

-- iL ^- rfl+s+tl, iTl+s+tl , Re(s) > 0, Re(t) > 0

3

exp[-xV-xV] expf-xV-xV]

1 1

e-x-y

1

_i_

|A|

r[

1" Bs-At

5^tA

L

-bs+at "1

, =^pt A

'

],

J'

A = aB-bA * 0

( A ) j (B-b)s+(a-A)t

4

B

exp[-max(xV,xV)l exp [-maxfxV y )l

^

(Bs-At)(at-bs) (Bs"At)(at"bs)

x r r(B-b)s+(a-A)t [(B-b)s+(a-A)t "1]

93

5

(l+x+y)""

6

[l+max(x,y)l

a-s-t I

£3,t

r

a

J

(1-x-y)""1

8

(l-max^y)]""1

g

1 1-x-y

1u

r

r

>

Re{s+t)

s, t,a Re(a)

_ a+s+t

> 0

1" 1+s+t

r(a) . L Reioi) > 0 "it" l L a+s+t

' s,t,l-s-t

r

1 l-max(x,y)

"1

— +s+t, — 2

1

-s-t

, Reis+t)

J

2

1+s+t,1-s-t

rf

— +s+t, — - s - t

11

Re{s+t)

a

L

1n

>

' 1+s+t,a-s-t-I

i

Reioc)

7

, Re(oi)

2

< 1

s,t,p--s-t -s-t

r

-P |1-x-y|

2

Re(s+t)

'

J

;+t,

L

— £

_U+p)__

s-t o L

7T r(p)C0s(p7T/2)

Re?(s+t

94

< Re(p ) < 1

<1

12

1

-f> |l-max(x,y)|

"it

r

r[

s,t,p-s-t

(1

" P ) +s+t,

(1+P)

2

-st _

2

n (p)C0S(p7T/2)

Ke(s+t) < /te(p) < 1

13

Ei(-x-y)

14

Ei["-max(x,y)

"

r(s)r(t) sTt

r(s+t) sTE

" s,t, - +s+t 1 15

erfc(vS^)

1+s+t

J

r( 1 +s+t) erfc(^max(x,y))

16

V^

1 7 i /

1R 1o

19

l0g

(

1

1 na( 1 1 + x+y

- r[s,tf-s-t|, / t e ( s + t ) < 1

)

- -^-r-r[l+s+t,-s-tl

log 6 1+ [ max(x,>

1

St

, Ke(s+t) < 1

1" S,t,S +t,l-S-t

1 I

n r

^h-^l

- +s+t,l+s+t,i -s-t L

2

2

/te(s+t) < 1 95

»

20

log 1-

s+t,1-s-t

1 max(x.y)'

st

l , . l . - +s+t, - -s-t L

2

2

/te(s+t) < 1

21

log(x+y) x+y-1

4s,t,s+t,1-s-t,1-s-tj,

Re(s+t)
22

- i ^ r|s+t,1+s+t,1-s-t,1-s-t],

log[max(x,y)3 max(x,y)-1

Re{s+t)
23

s,t,s+t,1-s-t,1-s-t

log(x+y) x+y+1

-n r - +s+t,

- -s-t

2

2

Re(s+t) < 1

s+t,1+s+t,1-s-t,1-s-t 24

log[max(x,y)3 max(x,y)+l

-7T r st

- +s+t,

- -s-t

2

2

s,t,s+t,1-s-t 25

n r

log 1x+y

- +s+t,1+s+t, - -s-t 2

'2

Ke(s+t) < 1 s+t,1-s-t 26

log 1-

max(x,y)'

st

- +s+t,

- -s-t

2

2

Re(s+t) < 1

27

log(x+y) x+y-1

]■

T|s,t,s+t,1-s-t,1-s-t Re(s+t) < 1 96

28

H>

-~ rTs+t.i+s+t.i-s-t,i-s

log[max(x,y)] max(x,y)-l

Ke(s+t) < 1

29

log(x+y) x+y+1

-TT

s , t , s + t , l - s - t , l - s - t "I

r

i +s+t,

i -s-t

Reis+t)

J

>

< 1

s+t,1+s+t,1-s-t,1-s-t 30

log[max(x,y)] max(x,y)+l

r

-7T

"it

i +s+t,

\

>

-s-t

/te(s+t) < 1

31

r s,t,s+t+ - , - -s-t

log l+v^+7

2

7i r 1

[

s+t,s+t+l,

1

2

1-s-t

|Re(s+t)| < i

32

log

r s+t+ -

l+AAnax(x,y) 1-Vmax(x,y)

"it

, ~ -s-t 1

2

r

|_

s+t, 1-s-t

J

'

|Re(s+t)| < i

33

34

lo v ^

fs , t , s + t , s + t +

I

+ y^l+x+>

—n

ZVxl

l o / m a x ( x > y ) + v'l+maxCx.y) 2vW(x,y)

i , 1 - s - t "I

1+s+t, 1+s+t

1 2v^st

[

1+s+t, 1+s+t Re{s+t) < 1

97

J

'

'

35

1

2 l+/lTx~+7

s,t,s+t,s+t,l-s-t "

1rf

> s+t+ i

,s+t+1 J

2

Re(s+t)

36

lo

^

2 l+v'l+maxtx,y)

r

"s+t,s+t,l-s-t

r

lit

vW(x,y)

< 1

s+t+

'

-

L

J

2

Re(s+t) < 1

37

1

|r

d I U 5D 111

•l+x+y

"s,t,s+t + -

, -

2

2

-s-t

S+t+ 1 Re(s+t) < -

38

1

3 r p ^ l v\

^

^l+max(x,y)

r|s+t+ i

, i -s-tl

Re(s+t)

39

< i 2

rs,t,s+t+y ■

1 s+t

r(p,x+y)

,

r r

> L

s+t Re(s+t+v) > 0

40

r[i>,max(x,y)]

41

J ^ )

-4 Hs+t+i/), st

r

Re(s+t+i>)

" S,t,S+t+

> 0

-£>

_ s+t, 1+-^--s-t _ --Re ( — )
98

42

J

(2v'max(x,y)]

s+t

"it

r c+t+ J^_

r

44

45

sin v ^

sT

vW(x,y)

r

v^ r

cos(2v6^)

2

2

Vtt

cos(2v'max(x,y))

it

[ S,t

KV( H

2

1

- "S-t J 2

, Re(s+t) < 2

r i+s+t r

irf 2

2

, Re(s+t) < -

L s+t

1

, Re(s+t) < 2 - -s-t 2

s,t,s+t+ — , s + t 2

L

2

>

S+t

Re(s+t) >

48

Re(s+t) < -

r - -s-t i

2

L

47

— )
_ s+t,l+s+t

L

46

J

's,t, I -s-t -1 2 , i/£ r

2

sin

»

1+ — -s-t 2 -Re(

43

i

J

\Re{v)\/2

I^r[s+t+^,s+t- ill,

K^(2^max(x,y)J

Z St

1

2

2J

Re(s +t) > \Re{v )|/2

99

F[;H

49 i

i

L

'

|s,t,a-s-t]

■[.b]

b-s-t

j

Re (s+t) < /te(a); b * 0, -1 , - 2 ,

l+s+t,a-s-t 50

P * -max(x.y) i iL I J

b-s-t Reist)

b * 0,-1

,-2,...

s,t,a-s-t,b-s-t

■[.%]

■ [v H

51

< Reia);

1 st '

c-s-t Reis+t)

< Re(a),

Reib);

c * 0,-1,-2,...

52

"

'

l+s+t,a-s-t,b-s-t

■[.%]

-max(x,y)

c-s-t

1 st

Rei s+t) < /te(a),/te(b); c *

0,-1,-2,

^s,t,s+t+l-b,a-s-t 55

* a,b;x+y

a,

a-b+1

Reib)-1

< /te(s+t) < fle(a)

l+s+t,s+t+l-b,a-s-t] 56

tf a,b;max(x,y)

st

a, Reib)-1

100

a-b+1 < Reis+t)

< Rein)

A:B:B' f (a):(b);(b>

(c),(d),(d')

57

-x>-y C:D;D'

(c):(d);(d');

(a),(b),(b')

(a)-s-t,(b)-s,(b')-t,s,t x r (c)-s-t, (d)-s, (d')-t

Finally, we give a representation of the G-function

(13.1) through

the sum of double hypergeometric Kampe de Feriet series (1.7) in the case m

= 0 (see also formula (14.1)) ,

*1

M

l

m

*2

M

2 '*3

M

,

(2).

): (a

f

(3) .

); (a

C0 (1) ): (Hl2));

3

A

) )

(P<3))

l+P(2V3)-Caa))

m

k k=l

(1),

(a

0 ,n :m ,n ;m ,n 1 2 2 3 3 x,y p ,q : p ,q ;p ,q

j

(a'")

j=l

1,n 2)

-p' -p! 3, > l +/ 3' 2, + p! 3) -(3 <1) ) i

,

n + l , p k

j

k

(2) (0 )*l,m - Uk( 2 , l l V k2 , - ( a l 2 ) ) l,n ' 2

X r (2)

2

(<x )

j

l,q

2

2

-p' >, 1 + p <2>-(^ >)

n +1, p 2 2

k

k

m +1 , q ■ 2 2

(13.9) (0 ( 3 ) )* -^(3),l^(3)-(a(3)) 1,«3 J J l>n3

X r

(a(3)) ^^3) n3+ l,P3 j

P : p

;P

xF q :q -l;q -l M M M

l

2

3

(3). (3), j V1>q3

l,p

'k 1

j

l,p

k 2

1-(P11,)1l,q + 8 k< 2 V 3j) : 1-(P<2')*l,q V 2 k' ; 1

2

101

l-(a ( 3 ) )

V3'; I,P3

J

p +p +n +n +m 2

(-1) '

*

2

p +p +n +n +m

2

x,

(-1) *

3

3

'

3

I-(P(3)); v . 3 , ; 1 , q

(a)

3

J

(p-q) is the abbreviation of the array of q-p+1 parameters p,q

(a)

(a)

= a ,a p

, ..., a ;

p+i

q

-a (l^k^p) denotes the array of p-1 parameters

1, p

k

(a)

-a = a , . . . , a k

i,p

and

p,q

1]

T The

l

,a

k-i

k+i

,...,a; p

is Slater's notation (4.33).

formula

(13.9)

holds

provided

the two following

sets of

conditions satisfy

(13.10)

p + p < q +q ,

p +p
3 ^1 3

*1

|arg(x)| < ^ (2^+ 2m^ \arg{y)\ J &

(13.11)

2 ^ - p ^ p ^ ^ ) ,

< ^ (2n + 2m + 2n -p -p -q -q ), n 2

1

3

3 *i *3 n

3

|arg(x)-arg(y) | < ^ (2m + 2m +2n +2n -p -p -q -q ). &

Here

(13.10) provides

&

2

2

the convergence

3

2

3 *2 *3 n 2

n

3

of the double Kampe de Feriet

series, which are involved in (13.9). The conditions

(13.11) give the

existence of G-function of two variables in the case ml = 0. We conclude by remarking that the representation (13.9) was obtained by R.P.Agarwal (1965) and B.L.Sharma the

last

inequality

corresponding

double

(1965), however, in both works there was omitted in (13.11). integral

As is known from

(13.1)

which

G-function of two variables can be divergent.

102

is used

Theorem

13.1,the

to define the

§14. The Double Kampe de Feriet hypergeometric series

J.Kampe de Feriet

(1921) initiated the study of the case B = B'and

D = D'of the double hypergeometric series

A:B;B' ( (a): (b);(b'); (14.1)

x,y (c):(d);(d');

C:D;D'

A T1 T

r

'

(a )

n

j m+n ' '

J

i - 1

C n=0 ~TT(c ) 1 ' j=l

B' B ~TT (b ) T~T (b')

j m+n

5 — 1

D FT

' ' j=l

j m ' '

J

in

X

m

n

y

\ = 1

D' (d ) ~TT (d') j m

' ' j=l

m! n!

j n

where (a) denotes the Pochhammer symbol defined by n

1,

if n = 0,

(a) = a(a+l). . . (a+n-1),

if n € IN = {1,2,3

>.

For the above double series H.M.Srivastava and M.C.Daoust (1972) deduced from much more general results (proved by them) that (i)

if A+B > C+D+1 and

A+B' > C+D'+l,

then

the

series

(14.1) diverges whenever x * 0 and y * 0; (ii)

if A+B = C+D+1 and

A+B' =

C+D'+l,

then

the

series

(14.1) converges absolutely, provided that

( (14.2)

max{Ixl,lyl> < 1 when A ^ C, i l/U-C)

|y| 1 / ( A _ C ) < 1 when A > C;

(iii) if A+B < C+D+1 and A+B'< C+D'+l, then the series (14.1) converges absolutely for all x,y € C. 103

It is understood (in each

situation)

that

no zeros

appear

in the

denominator of (14.1). In case (ii) above, the double series (14.1) also converges when x and y lie on the boundaries of the regions described by (14.2) provided some additional constraints, which were found recently by Nguyen Thanh Hai, O.I.Marichev and H.M.Srivastava

(1992) are imposed also Nguyen

Thanh Hai (1990a)). Our

solutions

of

the

convergence

problem

for

the

double

hypergeometric series (14.1) are contained in Theorems 14.1, 14.2 and 14.3 bellow. Theorem 14.1. Let A+B = C+D+l, A+B' = C+D'+l, and A = C. Then

the

series A:B+1;B'+1 f (a):(b);(b'); (14.3)

x.y A: B ; B*

(i)

converges

[ (c): (d); (d' );

absolutely A

when |x| = 1 and |y| = 1, if and only B+l

A

B

j=i

j=i

A = Re

if

< o, L

j=i

A

j=i

B'+l

B'

8 = Re

< o, j=i

j=i

j=i

j=i

and

A

B+l

j=i

j=i

B'+l

B

B*

j=i

j=i

c = Re

(ii)

< o,

converges

conditionally

if

A < 1, 5 < 1,

j=i

j=i

when |x| = 1 and |y|=l (x * 1; y * 1), and 104

c < 2;

(Hi)

diverges

when |x| = 1 and |y| = 1, if

at

least

one of

the

following three conditions does not hold true: and

A < 1, 5 < 1

c < 2.

Theorem 14.2. Let A+B = C+D+l, A+B' = C+D'+l, and C-A = k > 0. Then the

series A : B+k+l;B'+k+l f (a):(b);(b'); x,y A+k: B ; B' (c):(d);(d');

(14.4)

(i) converges absolutely when |x| =1 and |y| = 1, if and only if A

B+k+1

j=i

j=i

A+k

B

A = Re

< o, j=i

j=i

and A

+

B'+k+l

B'

c

& I ; -Z r &

5 = Re

j=i

(ii)

A+k

b

j=i

converges

conditionally

if

A < 1,

j=i

< 0;

j=i

when |x| = 1 and | y | = l (x * 1; y * 1),

and 5 < 1.

Theorem 14.3. Let A+B = C+D+l, A+B' = C+D'+l, and A-C = k > 0. Then the

series A+k:B+l;B'+l ( (a):(b);(b') ;

(14.5)

x,y A :B+k;B'+k

(c):(d);(d');

converges absolutely when |*|1/k

(14.6)

+

|y| 1/k = 1

(x * 0; y * 0)

if A+k (14.7)

e = Re

B+l

B'+l

A

B+k

B'+k

& 1 v I ;- I i - 1 v I d ; j=i

+

j=i

b

j=i

j=i

105

c

j=i

j=i

+k < 1.

Proof of Theorem 14.1. Case A

m

(i).

Denoting

the

general

term

of

the

series

(14.3)

n

x y , and making use of the familiar asymptotic estimate:

T(a+n)

(14.8)

Re(a-b)

,

.

we have

A FT 1

l A

I J=l A

m nI

A x y

=

B+l B'+l F T (b ) F T (b')

(a )

j m+n ' ' J=l B

"pj(c) 1

' j=l

j m ' ' j n J=l B'

m X

n y

7 7 (d.) 7 7 (<*•.)

j m+n

' ' j=l

(C ),(d ),(d') J J J

j m

' ' J=l

j n

a +m+n,...., a +m+n 1

|x| m |y| n

(a ),(b ),(b') J J J

A

, c +m+n, L

c +m+n

l

A

(14.9) b +m,....,b +m,b 1

B

+m

.,b',+n,b', +n B B+l

d'+n,..

.,d',+n,1+n

l

d +m,.....d +m,1+m 1

b'+n,.,

B+l

l

B

, .a /3-i 3T-i H (m+n) m n

B

TT f

where H

is a constant,

(m

> co; n

|x| = |y| = 1,

B

(14.10)

a =

Re L

D

B'

D'

j=l

j=l

Y*rYc\,v--Re\Y*i-Y«3[*--"e\Y?]- Id; j=i

j=l

j=i

j=l

J

L

C,D,and D'(and A,B, and B') being always specified in the context, and

r(a )...T(a ) i

b , . . . J,b

• l

p

is Slater's notation.

106

p_

r(b )...r(b ) i

p

by

To prove Case (i) of Theorem 14.1, it is sufficient to apply the following result.

Lemma 14.1. Let

a,/3,y e IR. Then

I

(14.11)

)

the

series

(m+n^'V""1

m,n=l converges

if

and only

if

a+|3 < 0, a+y < 0, and a+ft+y < 0.

Proof. Let us fix m = 1 (n = 1) and obtain the condition a+y < 0 (a+/3 < 0 ) . Hence 2a+/3+? < 0; and if a ^ 0, then *+($+? < 0.

In the case when a < 0, we have

(14.12)

>

(m+n) m

n

)

=

m,n=l

(m+n) m

+ )

n

m>:n^l

(m+n) m

n

,

l^m
and

(m+n) m'

n

=

m

n

1+ -

.

I mJ Since m ^ n, the hypothesis a < 0 implies that

a 2

Now

if we denote

"

£

(

1+

i)

the simultaneous

< 1.

convergence

or divergence

of

two

series by the equivalence symbol =, then the following relations are easily verified: co

)

(14.13)

i>n^l

mi

oo

im+nj m

n

=

>

oo

(.m+nj

rn^n^l

n

-

)

m

,

m

m=l ^

107

\

> n n=l

'

( lPct+0+y-i i^m"^"1 m=l

I

(y ( y > 00)).,

oo

« * JI

Vm^^logdn) V ma+/ log(n) m=l m=l oo a+8-i °° m

~ 1 I I I

E

I

[ m a+ ^ _1

(y (y == 00), ),

(y < 0 ) . (y < 0 ) .

m=i

Hence the condition a+/3+y < 0 follows, and the proof of the necessity of Lemma 14.1 is completed.

Next we observe that if a £ 0, then mmaa**fifi'1'n1n7r7r**0La""11..

3r_1 a Cm+n)am^"1n3r ^^

Thus the convergence of the series (14.11) follows from the conditions a+£ < 0 and a+y < 0. On the other hand, if a < 0, then the convergence of the first series on the right-hand side of (14.12) follows from the relationship

(14.13).

The

convergence

of

the

second

right-hand side of (14.12) can be proved similarly.

Case(ii). Casedi).

on

the

We shall need the following result in this case.

Lemma 14.2. Let

a,/3,y e R, and = (m+iO^-n-1.

e

Then, for Then, for

series

■

the the m,

equalities: equalities:

lim e n—Xx> lim e

m, n — X »

to hold true,

mn mn

it

= lim lim c = m—>oo lim n—>oo lim c m—X»

is necessary

= lim lim e m—X» e = n—>oo lim lim

n—X»

n—X»

and sufficient

m—Xx>

that

en en

=0 =0

a+|3+y < 2, a+/3 < 1, and a+y < 1. a+^+y The proof of Lemma 14.2 is easy, and we omit the details involved.

Setting A x m y n = u v mn

mn mn

, where v mn

108

=

x m y n and

A

(14.14)

u

j[= 1 '

-

j

7 7 (b.) 7 7 (b'j m+n j =', 1 '

A

j mJ =[ 1 '

B '

j

l e t us use the conditions

j

1 n

B'

1

m!

n!

7 7 (d ) 7 7 (d'j

77(c) 1

B'+l

B+l

7 7 (a.)

m+n

'

'

|x|=|y|

j m '

=1

'

j

n

(x * 1; y * 1 ) ,

a+0 < 1, a+y < 1, and a+/3+r < 2 . Then i t

i s r e a d i l y seen

that in

1. The p a r t i a l

2.

{u

3.

the

sums S

mn

=

ii

)

)

L

v

L

ij

a r e bounded;

> c o n v e r g e s u n i f o r m l y t o z e r o when m —> oo and n —> oo; mn

series f |u

L ' m,0 m=0

- u

I,

m+1,01

f

lu

L ' 0,n n=0

- u

I

0,n+l',

oo

and

) | u - u

U m, n = 0

' mn

m+l,n

- u

m,n+l

+u

I

m+l,n+l '

are convergent. By appealing now to Theorem 1.1.3 of A.Yanushauskas the conditional convergence of the series

(1980), we obtain

(14.3) under the specified

conditions. Case

(iii).

It

is

obvious

that

the

condition

mn m, n — > oo

necessary for the convergence of the double sequence {u

>. Therefore,

mn

the validity of the assertion of Theorem 14.1 in Case (iii) follows from the relationship (14.9) and Lemma 14.2.

■

Proof of Theorem 14.2.

Case

(i).

With the help of the asymptotic estimate (14.8), we find

for the general term of the series (14.4) that

109

,I

A

1 1 1 1

(14.15)

j j m m ' ' ' '

—— B

—— B'(d B'

j j n n

X X

rr«v.-» r r «v.rT (d ; }} » rr«v.-» r r «v.rT ; »

1

j=

i

J=

I

j=

yy

m! n! m!

i

.a+k 6-k-i y-k-i ii iimm i i (( nf n f m!n! m!n! 11 TT , r, ~ H (m+n) m nn xx yy r, where H

. -

B'+k+l

T T (b.) (M T FT <»f>.)•.) T~f 7 (b

j j m+n m+n ' ' ''

|B |B x"y"| x"y"| == MM^^ mn mn A+k A+k

(14.15)

B+k+1

T7 7 (a.)

is a constant, and a,0,* are defined

,, .. Cm —> °°; oo; n —> °°)» oo), (m

by

(14.10) with

the

2

numbers of parameters appropriately specified as in (14.15).

numbers of parameters a p p r o p r i a t e l y s p e c i f i e d as in (14.15). By Stirling's formula:

By S t i r l i n g ' s formula:

. . ,,

we have we have

, , , , ,— —r.~ m!n!

/=— .. k+1/2 k+l/2 pr-

k! k! ~ ~ v27i v27i k k

e e

m+l/2 m+l/2

n+1/2 n+1/2

V^TT pr- m

-k -k

,...

,,

(k (k —> —> oo), oo), (m

oo;

_>

n

oo).

_>

(m —> co; n —> oo).

n

2 (m+n)! ((m+n) m+n)m+n+1/— ( i ^! Then, for Ixl = 1 and lyl = 1, we find from (14.15) that Then, for |x| = 1 and |y| = 1, we find from (14.15) that j-

,„ ,„ |B

m

n

-i

K

m n, ,_ .kTT . .a+k/2 /3-k/2-l y-k/2-1 j-— m n -i k m n, ,_ .kHTT (m+n) . .a+k/2m /3-k/2-ln y-k/2-1 — xy | ~ (271) (m+n) m + n mn 2

|B xy R| H ~B' (271) H2(m+n) = mn(2TT) .

m

(m+n) m + n

n

2 mn

= (2rr)kH B' . Furthermore, for sufficiently large m and n, 2 mn Furthermore, for sufficiently large m and n, ro n n m m n

, ,

= =

.m+n .m+n

'•*"'

Hence Hence we we obtain obtain

1

I-

1

//

y,m ^m

f,

^n xn

M) M) B' B'

(14.16)

..

\-

mn mn

< <

1

<< _L _L — —

(p R+). (p €€ R+).

pp p p

" ■

a+k/2 1 2r (m+n)a+k/2 m^"k/2"1_ "pkn "k/2"11~pk "pk. (m+n) m^~k/2~ pkn 2r ~k/2~

In accordance with Lemma 14.1, the double series with a general term of the type (14.16) converges for sufficiently large values of p. So, by taking

into

account

the

conditions

conclude that the series 110

A = a+/3 < 0

and

S = a+r < 0, 6

we

I

m,n=l converges. The sufficiency of the conditions in Case (i) of Theorem 14.2 is proved.

The necessity will become evident if we fix m = 0 (n = 0) in the series (14.4).

Case

(ii) of Theorem

Theorem 14.1.

14.2 can be proved

in the same way as in

■

Proof of Theorem 14.3. Let us consider the general term of the series (14.5) (m+n)! , „ .

. „ .

(14.17)

i _ 1

m Hi

C x y mn

TT

'

= HC

/-

xtt+k

~ H (m+n) 3

/3+k-l

3f+k-l

na

m'

i

i

x

m

i

i

n

y

xT y ".

3 mn1

' ' '

By the principle of mathematical induction, it is not difficult to obtain the inequality: -.k (m+n)!

(14.18)

[k(m+n)]! (km)!(kn)!

Now c h o o s e M € IN so

that

kM+/3+k-l > 0 Then we g e t t h e

and

kM+y+k-1 > 0.

inequality: kM + 8 + k - l

kM+^+k-l

m n 2kM+2k + |3+3r-2 (m+n)

Hence

111

< 1.

y c'mn x

I m I

in

y

zL

i,n=l 00 r—«t

kM k -- Il K P l ++ p/ 3++K

_

) (m+n) / *—■ „ m,n=l

,

(m+n)!

m n

I )

.2kM+2k + j

(m+n)

m+n)

<

(m+n)

x

y

(m!n!)

v 2M

a+/3+3-+k-2

(m+n)

m n

m,n=l nn,n= 00

(14.19)

^kM+^+k-1

-I

(m+n)

*

-l

(m+n)!

(m+n+2M)!

a+p+^+k-2

x

*

y

(m! n! )

|x|m|y|n

(m+M)!(n+M)!

m,n=l

| xy | '"

a+|3+3r+k-2 (m+n)

>

m,n=l

(

[k(m+n+2M) ] ! [k(m+M)]![k(n+M)]!

ky K/

xk(M+m) N mn+nu

k, •, K/

Nk(M+n) xH

00

xy|-"^l-^*k-a(^

k>

+

xk(2M+l)

{^f)'

1=2 00

=

xy

M^

i a + / 3 + r+ k-2

js.nce ^

+

^

=^

The last series converges if a+/3+r+k < 1. This evidently completes the proof of Theorem 14.3.

■

For an illustration of the above Theorems 14.1 - 14.3 we consider the well-known Appell series F

- F

(see A.Erdelyi at al.(1953), V.l)

which are special cases of the Kampe de Feriet series (14.1). 112

Consequence 14.1. The first

Appell 1:1; 1

(14.20)

F (a.p.lTjrjx.y) =

series

f a:0;/T;

F

x,y 1:0;0

y:-

(a)m+n ^ O )m (£') n

C O converges absolutely

when |x| = |y| = 1, if and oniy if

< 0,

Re(a+^'-r) < 0

(iij converges conditionally

diverges

when

|x| = 1 and

following

three

conditions

Re{oc+f5-v) * 1,

and Ke(a+/3+0* -y) < 0;

when |x| = |y| = 1 (x*l, y*l), if Re{oc+(S*-y)

Ke(a+/3-r) < 1, (Hi)

y

(y) m+n

m,n=0

Re{a+p-v)

x

< 1

and

Re(a+^+j3' -y) < 2;

|y| = 1, if

does not hold

Ke(a+£'-y) * 1

and

at

least

one

of

the

true: Re(a+/3+/3' -y) < 2;

Remark 14.1. In the literature, see, for example, P.Appell, J. Kampe de Feriet

(1926), A.Erdelyi et al. (1953,v.l), there is the following

formula (14.21)

Fta.p^'jy;!,!) = T r-a, y-/3-/T

(14.22)

However,

/te(a+0+0'-r) < 0.

provided

the

condition

(14.22)

is

given

the

series

F (a,j3,j3* ;y; 1,1) is not obligatory convergent, i.e., the limit N

lim M , N — > co

does not always exist

N

II-

^ ^ J . «

m ! n !

m+n

m=0 n=0 (for example, when a = y, £ = 1/2, £*= -3/2).

Therefore the formula (14.21) means 113

(a) A (0) (0») m+n m n

F (a,/3,/3' ;y;l,l) = lim k—xx)

( y m+n )^

m+n=k

m!n!

k y,y-a-/3-/3' lim k—x»

y-a, y-/3-/3' J

j=0

provided (14.22) holds true.

Consequence 14.2. The third

Appell

series

0:2;2 f -:a,a' ; 0;0' ;

=F

F (a,a;/3,/r ;y;x,y)

x,y 1:0:0

y:

(a) (a') (3) O ' ) m n m n

x m yn m! n!

Cr). m,n=0

(i)

converges

absolutely

when

|x| = |y| = 1, if

and only

if

fie(a+0-y) < 0 and Re(a'+ 0'-y) < 0; (ii)

converges

conditionally

when |x| = |y| = 1 (x * 1, y * 1)

Re(a+0-y) < 1 and Re(a'+ 0'-y) < 1.

Consequence 14.3. The fourth

Appell

series

2:0:0 f

a,0:-; -;

0:1;1

-

F (a,0;y,y' ;x,y) = f

x,y

n m,n=0

converges

absolutely

:y;y

(a) x (/3) ^ m+n m+n

x m y11 ^

(y) (y ) m n

m! n!

when v|x| + v|y| = l,(x * 0; y * 0) if 114

/teCa+p-r-y') < -l. Consequence 14.4. The second Appell 1: 1; 1 F (a,|3;/3' ;y,y' ;x,y) = P 0:1;1

series

a :0;0';

1 x,y

(14.23) (a) _,_ (0) O ' ) m+n m n

x m y11

(y) (y') m n

m! n!

_ m,n=0 converges absolutely

when |x|+|y| = l,(x * 0; y * 0), if

(14.24)

Re(a+0+0'-*-*' ) < 0.

Here Consequences from Theorem

14.1, 14.2 and 14.3 - 14.4 follow

14.1, 14.2 and 14.3 respectively.

immediately

It is interesting to

remark that (14.24) is necessary and sufficient for the convergence of the series F

(14.23).

Theorem 14.4. For the second Appell following

three statements

1) The series

that

series

(14.23) the

equivalent

(14.23) converges

y = 1-a, where 2) The series

are

absolutely

at

least

for

x = a,

a € R, 0 < a < 1.

(14.23) converges

absolutely

for

all

x,y e C such

|x| + |y| =s 1, x * 0, y * 0.

3) The inequality

(14.24) holds

true.

Proof. First note that the chain 3) =» 2) => 1) is evident. We must prove that the statement 3) follows from 1). Since for the series F ^

2

(14.23) the following symmetry property holds valid

F (a,|3;0';y,y';x,y) = F (a,0f;0;y*,y ;y,x),

115

then we can consider 1/2 ^ a < 1 without emissing a community. In this case for the second Appell series F

we write the asymptotic representation

obtained by M.Saigo, O.I.Marichev and Nguyen Thanh Hai (1989)

F (<x,/3;/3' ;y,y' ;a-p,l-a-5) ( a, l + a - y , y ' -/3'

y,y-a-/3 F y-a,y-/3_

J 3 "^

3 2[

l+a+0-y,

y'

a-1 a

y , y ; a + / 3 - y , y + y ' -a-|3 -a)y"a^?

r

a , / 3 , y ' - / 3 ' , y+y' -a-0

(14.25) i-/3,y-/3,y+y'-a-/3-/3

a-1 a

x F 3 2

1 -a-/3+y, y+y' -a-/3

y,y', a+/3+/3'-y-y' (p+5)y+y -a-

a ^ ( 1 _ a ) /3'-y' +

Q(]

a,/3,y

where p F

+0, 5 ^ + 0 , y-a-/3, y+y'-a-/3-/3' are not equal to integers, and

is a Clausenian function (A.Erdelyi et al. (1953), V.l).

3 2

Now let in (14.25) p and 5 —> +0. Then by second Abel's theorem (see, for example, A.Yanushauskas (1980)) the left part of (14.25) tends to a finite

limit F (a,/3;/3'; y, y'; a, 1-a) since

(from

the statement

1 this

series converges absolutely for the point (a,1-a)). On the other hand 'x+'x' — oc—fi- 8'

the right part of (14.25) contains the expression (p+5)

' ' , that

is why it converges to a finite limit if and only if Ke(<x+/3+£' -y-y' ) < 0. The proof of Theorem 14.4 is completed by following two notes. At first, here the conditions y-a-/3, y+y'-a-/3-/3' = 0,-1,-2,

are

not essential.

At second, in the case a = 1/2 the formula is also true, if we set 116

(a,b,c; F

fa,b,c;

) -1

3 2

lim

F , 3 2

d,e;

d,e:

z -^-1 Since the Clausenian function

F

is analytic near the point z

limit always exists (although the Clausenian series

-1, this

F (z) converges for

z = -1 if and only if Re(a+b+c-d-e) < - 1 ) . Finally, we note that the Kampe de Feriet series can be represented by the G-function (13.1) of two variables as follows

A:B;B' f (a):(b);(b'); (14.26)

p

(c),(d),(d>) -x,-y

C:D;D'

(c): (d);(d');

(a),(b),(b')

(27Ti)

(a)-s-t, (b)-s, (b' )-t,s,t' x y dsdt (c)-s-t,(d)-s,(d')-t L L t s

(c),(d),(d')

0,A: 1,B ; 1,B'

(a),(b),(b')

A,C:B,D+1;B',D'+1

l-(a): l-(b);

l-(b')

'

x,y l-(c):0,l-(d); 0, l-(d')

where (14.27)

A+B < C+D+l,

A+B' < C+D'+l

and f |arg(x)| < ^ (A+B-C-D), (14.28)

Urg(y)| < \

(A+B'-C-D'),

Urg(x)-arg(y)| < | (B+B'+2-D-D').

(14.28a) (14.28b) (14.28c)

Here (14.27) provide the convergence of the Kampe de Feriet series (14.1), but conditions (14.28) are required for the convergence

117

of

the

double contour integral in (14.26). The relation (14.26) can immediately follow from the series representation of the G-function (13.9). We

conclude

by

remarking

that

provided

(14.28) holds

true

the )

A:B;B'f(a):(b);(b*); representation

(14.26) gives a meaning to f

-x,-y

C:D;D' [(c): (d); (cT); in the case when (14.27) does not hold true.

118

J

CHAPTER III. ONE-DIMENSIONAL H-TRANSFORM AND ITS COMPOSITION STRUCTURE §15. Spaces UJf^L) and 3JT1 (L)

In this Chapter we will consider a generalization of

the

next

integral Mellin type convolution transform

(15.1)

(Kf)(x)

k(xu)f(u)du,

x > 0,

o where k(x) in general case is Fox*s H-function of one variable (1.5). The transform (15.1) in various forms is considered by many mathemati­ cians

in

traditional

Mellin-Parseval

spaces

Lp (1 ^ p ^ +»). Here

formula we can represent

with

the transform

the

help

of

(15.1) in the

next type r+ioo

(15.2)

k (s)f (l-s)x"sds,

(Kf)(x) = ^ I r-iw

where f (s) denotes the Mellin transform of functions f(x) (see, for example, E.C.Titchmarch (1937), O.I.Marichev (1983))

(15.3)

f (s)

(y-ioo,y+im)

f(x)xs~ dx

is some vertical contour in the complex plane s. Thus the

transform (15.1) can be studied with the aid of asymptotic

estimations

of the functions k (s) and f (1-s) on the contour (y-ioo, y+ioo). It not

difficult

defined

on

to notice the

replacement

of

convolution

form

that

same

line.

variable

we

if

y = 1/2,

Putting can

write

as follows 119

then

these

f (x) = x f(x transform

functions )

with

(15.1)

in

is are

simple Mellin

(15.4)

1 2ni

k(-)f(u)-

(Kf^Hx) o

where
k (s)f (s)x ds,
= 1/2}. Mainly we will consider transforms of

type form (15.4) due to its large convenience for our further studies. Thus the behaviour of the functions k (s) and f (s) on the contour cr forces us to give the definition of the special functional space 3JT (L) introduced by Vu Kim Tuan, O.I.Marichev and S.B.Yakubovich (1986). As it is showed

below,

this

space

is very

convenient

for

the

studies of

transform (15.4).

Definition 15.1. Denote by 311" (L) the space of functions f(x), x € (0,+co),

representable

by

inverse

Mellin

transform

functions f (s) € L (o*) = L(tr) on the contour
(15.5)

The

f(x) = CTf1/ f*(s); x j = ^ j

space

3J1~ (L)

with

the

usual

of

€ C, Re(s)

integrable = 1/2}:

f (s)x ds.

operations

of

addition

and

multiplication by scalar is a linear vector space. If the norm in 5Jl~ (L) is introduced by the formula

(15.6)

|f (l/2+it)|dt,

m (D then the space 3H~ (L) is Banach.

Now we consider the main properties of the space 3JT (L).

1) f(x) € aif^L) if and only if x"1f(x"1) € UJT^L). This property exists as a result of the fact that the functions f(x) and x~ f(x~ ) are the inverse Mellin transforms of the functions f (s) and f (1-s) respectively, which simultaneously belong or not to

L((r). 1/2

-1

2) If fix)

€ UK

(L) then x

f(x) is bounded uniformly, continuous

120

on ((0, on 0 , +00) +oo) and and ffurthermore urthermore x x TW: _

__^___J...

r_ I I -

u c

,

ff(x) (x) =o o(l) ( l ) when x -> -» 0o0o a and nd x x-> - >0. 0. r_ _ .«__

■|-V"i*=» ■f_V-i*=»

R i p m a n n - 1 (aV^occniP

1 ^mma

f_..

F* C m a r cr hh 1937) 1 Q^7 1 E C Ti T i ttcrhh m

e JJJf'dJ J f ' d J then x1/2f(x)g(x) e L). 3) If f(x), g(x) e e ^VC(1{L). This follows from the fact that x1/2f(x)g(x) is the inverse Mellin transform of the function ^

f*(T)g*(s-T+l/2)dx

which belongs to L(
by Fubini theorem. Here g(x) = aOJf^g'Cs); n ^ j g ' C s ) ; xj.

00

4) Let f(x) e JJf'CL) J J f ' d J and x"1/2g(x) e L(R L(R+). Then |g(u)f(jS)*H [ g ( u ) f ( ^ belongs belongs + ). Then 0

t o nf'cu. toJJf'CL).

In fact by the property of the Mellin convolution this integral is the inverse Mellin transform of the function f*(s)g*(s) and since f*(s) € L(cr) and g*(s) belongs to space of essential bounded functions on
As is known the Mellin products

of gamma-functions

transform

of H-function

and according

is the ratio of

to asymptotic

of gamma-

function this ratio has power-exponential behavior on the contour o\ Therefore it is necessary to take into consideration this fact in the spaces of type 3Jf * (L). Definition 15.2. Let c,y e K be such that (15.7)

2sgn(c) + sgn(^) ^ 0.

Denote by JJf1 (L) the space of functions f(x), x e (0,+co), representable by the inverse Mellin transform (15.5), where f*(s)|s | 7en° ' I m S U L(cr). Note that f*(s)|s|V c | I / n s | e L(cr) if and only if f*(s)|s|V C |s' e L( 0, y € R, 121

or c = 0, y > 0, which is equivalent to (15.7). The space 3JT

(L) is Banach with norm

enc\lms\

(15.8)

,sV(s)ds|.

ffl (L) c,7

It is obvious that the space UH

(L) in case c = 0, y = 0 coincides

with the space 3H (L). The following theorem takes place.

Theorem 15.1. For the

set

of spaces

(L) the

3fl

following

statement

c,y

is

true (L) c m'1

UJf1

(15.9) if and oniy

(L)

if 2sgn(c -c) + sgn(y -y) >: 0.

(15.10)

Proof.

Suppose

that

inequality

(15.10)

holds

valid.

Then

from

(15.8) we have

l f l -a m

e

Tic | J O T S |

I

y

*

^,

s f (s)ds

i

(D f

(15.11)

TIC | JOTS |

e

f :£ C

7i(c-c ) I Ims |

y

Is *f (s)e

TIC

I Jms I

e

*

y # |s *f (s)dsl =

s

y-y

^sl

CIIfII 3R

(D

c ,y I I

where constant C is defined by the inequality (15.10)

(

TI(C-C

) \Ims\

e

y-y

|s|

I < +oo.

Now suppose that inclusion (15.9) holds. Then f(x) e W f 1 (L) c lfl'1 (L) c .r c,y -TIC |J/ns| -y -l-e # X a if f (s) = e I s| , e > 0. For this f(x) from (15.8) we 122

have .,

e

m

,

Tic Ims

|

-7tc \Ims\

y

1

s e

i

-y

s

i

-l-c

_, i

ds

(D 7r(c-c ) | Ims | e

Hence for all e > 0

Is

y-y - l - e d s l < +oo,

e > 0.

we get

2sgn(c -c) + sgn(9r -y+e)

> 0,

which is equivalent to (15.10). This completes the proof of Theorem 15.1. Remark 15.1. Note that 3JT1 c

and y

= y.

l

Then for yt

y , e, 1

1

(L) s 3JT1 (L) if and only if c = c

1»y1

c

J

c,v

€ (R, moreover e > 0, e

l

l

> 0, we have

the following inclusions 1 an"1 (D = m' (u 0,0

D

1 m' 0 ,C

( D D JJT1 C

(15.12) 3JT 1

D

e ,y +e

D im" 1

(L)

c +e ,y

l i

From t h e l a s t p r o p e r t i e s f o r 3)1

(D

,2T 1 1

(L).

l

(L) i t

i s not d i f f i c u l t

to

obtain

c,y

the following result. Consequence 15.1. If pairs

(c ty

) and

(c ,y m'1

c ,y

where

the pair

(c,y)

inequality ),

m as

1

c ,y

for

some

c,y

follows < c,

1

, if c

(c,r) =

2

< c,

2

(c ,min(3r ,3r )) 1

also

(L) s 3JT1 (L),

, if c

1

valid

then

(L) U

is defined

(15.7) holds

2

1

, if c = c . 1

2

To establish further properties of the space 3fl following auxiliary Lemmas. 123

(L) we need the

Lemma 15.1. Let all

k k

also

r ( \ 2|t| 3 eexp x p Rjs+tl+Rjsl+R^tl Rjs+t|+Rjs|+R 3 |t| . the vertical

linss iines with

1/2, i.e. , (cr x cr ) = j(s,t) e C 2 , /Res)) t e s ) ) = Re(t) Reit)

if

sup S(s,t) < +oo for sup

and only

(15.15)

the

real

parts

= 1/2 V. V Then Then

(s,t) e (cr x cr ) s

t

if [

(15.15)

and

r

c
(15.14)

numbers

k k

r r ~(s,t) = |s+t| l\s\ S(s,t)

(15.13) Let

R , r , k = 1,2,3, be real

2sgn(Rk+R.) + sgn(r sgn(r k+r.) +r.) < * 0, 0, r3

3

J r^

f

1

1]

Y \

k,j=l

v

J

{

k *j k,j=i

v

J

{

sgn(Rkk+R.) +R.) + + 2sgn 2sgn sgn(R j

j

k,j== 1,2,3; 1,2,3; kk ** j; j; k,j

.

fr r V

\ s * 0. 0

k Ik=i J =i k

U

J

k*j

This Lemma follows immediately from Lemma 11.4.

Lemma 15.2. Let

S(s,t) be defnned

by

(15.13). Then the

ineegral

I H(s,t)|str j S ( s > t ) | s t r 1 1--CGddsdt sdt

(15.16)

cr cr t s

convergss

(15.17) (15.17)

for all

c > 0 if

and only

if

f (

2sgn(R +R ) + sgn(r +r ) * 0, 0,

iI

2sgn(R +R ) + sgn(r +r -1) * 0, 2 3 2 3 3 2 3 2 3

I

II v v

l k 1 k

1 1

k k

f33 ]

3

V sgn(Rk+Rj ) + 2sgn £ k r V £ k r ^- sgn(Rk+Rj ) + 2sgnlk=i

^-

[k=i v v

k,j=l k,j=l

k*j

124

j j

J J

k = 2,3; k 2,3;

ss 0. s0

following

Proof.

Applying

Lemma

11.2 to integral

(15.16),

we have

that

it

converges if and only if

( 2sgn(R +R ) + sgn(r +r -e) < 0, 3 k ° 3 k 2sgn(R +R ) + sgn(r +r -l-2e) < 0, 1 I 1 22 *1 22

(15.18)

kk == 1,2; 1,2;

I 2maxJ 2 m a x | sgn(R sgn(R +R +R ) ) + + sgn(R sgn(R +R +R ), ) , sgn(R sgn(R +R +R ) ) + + sgn(R sgn(R +R +R ), ), 2 2 2 3 \ sgn(R +R ) + sgn(R +R )I + sgn(r +r +r -2c) < 0 ^ ! 3 3 1 1 1 1 2 2 J1 1 1 2 2 3 3 for all all e > 0. From three first inequalities of (15.18) (15.18) it follows that max{ R +R ,R +R ,R +R } ^ 0. max{

Hence

it

is

easily

verified

that

if

A u 2 l2h3 u 3 l3st / 1 inequality • ,-4. of , , • equivalent - , . 4 . the U i C ,o, is e-> +0,t2en (15.18) to last one e +0,then the last inequality of (15.18) equivalentttto (15.17). the last Tois one e-^ (15.17). Consequently, (i5.1o) i1 also is squivalent e(15.17). i1 also oompletes theConsequently, proof of lemma(i5.1o) 15.2. ■ oompletes the proof of lemma 15.2. Note

that

(15.17) (15.17)

follows

squivalent

tt

(15.17).

Tois

■

immediately

from

(15.15). (15.15).

Further

we

need the the following result in our our further studies.

Lemma

15.3. 15.3. Let

S(s,t) S(s,t) be

defined defined

by

(15.13) (15.13) and

R ,r , r ,, kk == 223 223 k

satisfy

the

following

(15.19) Then integral

k

inequauaty inequality 2sgn(R +R ) + sgn(r +r ) ^ 0 . 2 3 2 3

inequality

(15.14) follows

from

the

convergence

of

the

double

(15.16).

Indeed,

it is sufficient

then conditions

to note that

if inequality

(15.19) holds,

(15.17) are equivalent to (15.15).

Here for the space OH" OH"1 (L) we give one main property which

is used

many times in our further studies.

Theorem 15.3. 15.3.

Let Let

1

f(x) G LL )) ,, g(x) G OH'1 (( g(x) € € Jn" Jn"1 ((D L ). . Then Then c c ,? c1 ,? ,? 1 2 2 1 1

function h ((xx)) = — x x h

( 1 D . £\J )

(15.20)

belongs

to

the

1/2

f(x)g(x) f(x)g(x)

space 125

the

5H"1

(15.21)

( L ) = m'1

3

i.e., the

pair

c

3

1

(c ,j

^

3

) is

( L ) U 3JT 1

,3T 1

2

defined

as

2

(L) U W"1 (L), (ci+c2)/2,3ri+r2

follows

3

r (ci>ri) (15.22)

(c ,r 3

(

) = i 3

'

,

if

h(x)

c

< c 2 c < c 2 1 1

if

W

(c^min

Moreover,

if

(yi,y2,yi+y2)),

e

5K * (L) f o r c,r m'1 (L), then ffl-1 (L) D W" 1 (L). c ,y c, y c ,2r 2 2 3 3

aii

if

<^ = c 2

f(x)

€

!JK * (L) c ,y

and

Proof. From Consequence 15.1 and relations of inclusions is

not

difficult

According

to

show

to Definition

that

(15.21)

is

15.2 of the space

equivalent

3ft

g(x)

€

(15.12) it

to

(15.22).

(L) we can represent

function h ( x ) (15.20) in the next form 1/2 (15.23)

h(x)

f*(s)g*(t)x

2 (2?ri• ^

s

\isdt,

t s where (rxcr = {(s,t) € C 2 , Re(s) = Jte(t) = — > . s t 2 By replacing of variable x = s + t - — we can write (15.23) as follows

(15.24)

where

h(x)

F(T)X

2rri

dx,

cr = {x € C, fte(x) = — > and x 2

(15.25)

F(x)

27Ti

f (x-t+—)g 2

(t)dt

for x € cr . X

Consequently, according to Definition 15.2, we have h(x) e 5H some pair

(c,^),

if F(x)|x| e

c

. <= L ( — - i o o , — + i o o ) .

tion (15.25) we get the inequality 126

Using

(L) for

c,r representa­

eTrc|T||T|9r|F(T)dT|

7Tc|s+t-l/2| ,

_1

l.y,

*

.

«

r n

,

,. ,

| s + t - — | a | f (s)g (t)dsdt|

e

2TT

cr cr

t s

Since f(x) € JJI (L), g(x) e OH ( L ) , then c ,y c ,y 2*2

1 1

wcjsl e

^ * |s I

wc lt|

f (s) € L(cr )

and

e

*

y . |t| g (t) e L(cr ) .

Hence it is evident that the least double integral converges if

(15.26)

sup

'Hc

e x p U c | s + t - - i - | _ c | s | - c |t| I I | s+t--|- | |s| * | t | 2
( s , t ) €(T x(T s t

')]'

It is not difficult to note that the last inequality is equivalent to

sup

y f ^ ~yi exp c|s+t|-c | s | —c |t| | s+t | |s|

( s , t ) €(T xCT s t

^

"r2 |t|

< co.

'

Hence by using Lemma 15.1 we get that (15.26) holds if and only if

2sgn(c-c ) + sgnt^-r ) < 0, k

2sgn(-c -c ) + sgn(-y -y

(15.27)

k = 1,2,

k

)

< 0,

sgn(c-c ) + sgn(c-c ) + sgn(-c -c ) + 2sgn(y—If

Since

the pairs

(c ,y ) , (c ,y ) are satisfied 1 1

2

-

y ) - 0.

(15.7),

then

is

2

easily verified that conditions (15.27) hold valid for c = c , y = y , where the1 pair (c ,y ) is defined by (15.22). Therefore, we obtain that h ( x ) € m' (L). c ,y 3 3 Now suppose

h(x) = x

1/2

that there is some pair

(c,y) such that the function

1

f(x)g(x) belongs to UJf (L) for all f(x) e 3JT1 c, y

m

(L). Then

c »y

we choose f(x), g(x) such that

V*2 127

1 1

(L), g(x) €

|s |

-TTC

f*(s)

= e

*

-y - l - e |s|

-7ic | t |

1

and

2

g (t) = e

-y - l - e |t|

2

In t h i s c a s e h ( x ) ( 1 5 . 2 0 ) b e l o n g s t o JJJf1 (L) f o r a l l e > 0 . Hence from Definition

1 5 . 2 and ( 1 5 . 2 4 ) - ( 1 5 . 2 5 )

we o b t a i n h ( T ) = F ( x ) f o r x e
Consequently,

+00

>

e n c | t | | i i y i uh* (rx )^d xI

— 2TT

X

1 2?r

exp[ir[c|s+t--i-|-ci|s|-c2|t|)] t s y

-y

-y

-i-e

x|s+t-— | |s| *|t|2 |st|

|dsdt|

for all e > 0.

Since here 2sgn(c ) + sgn(? ) > 0, then 2sgn(-c -c ) + sgnt-jr -y ) * 0, applying Lemma 15.3 we get that the last double integral converges if and only if inequality (15.26) holds. Hence we again obtain conditions (15.27). We shall prove that in this case (15.28)

JJTf1 (L) D UK"1

c*

(L) U Ulf1

c ^

(L) U 3JT1

c2,r2

(L).

( V c 2 )/ 2 , V y 2

Indeed, applying Theorem 15.1 to the two first inequalities of (15.27) we obtain (15.29)

3JT1 (L) 3 3JT1

c,y

c ,r

(L) U JJf 1

c ,y 2

1 1

(L). 2

and, moreover, c ^ min(c , c ). Hence, for the case c < min(c ,c ) from 1 2

1 2

relations of inclusions (15.12) it follows that 3JT1 (L) z> m ' 1 (L). car ( V c 2 )/ 2 ,r i + y 2 Let now c = min(c , c ). Then note that min(c , c ) < (c +c )/2, if

(15.30)

1 2

1 2

1 2

c„* c . Hence in this case inclusion (15.28) follows from (15.29). We 1 2 must prove that in the case c = min(c , c ) = c = c inclusion (15.28) 1 2

is also valid. 128

1

2

Indeed, in this case from the last inequality of (15.27) we get 2sgn(y-y -y ) ^ sgn(c -c) + sgn(c -c) + sgn(c +c ) = sgn(c +c ) < 1, °

1

2

° 1 1

°

2 2

°

&

1 12 2

&

1

2

i.e., i.e., y y < < y y +7 +7 . . Hence Hence using using relations relations (15.12) (15.12) we we obtain obtain that that (15.28) (15.28) holds. holds. Finally, we

see

that

in both cases

inclusion

(15.28) is always

(15.21) it means that JJf1 (L) D 3JTl (L). cy c33,3r ,y3 This completes the proof of Theorem 15.3. ■

valid.

In accordance with

Consequence 15.2. If

both

f(x),g(x) € 3Jt _1* (L), (L), then then xx

1/2 1/2

f(x)g(x) f(x)g(x)

c,y

3JT1 € 3n

c,min(y,2y)

(L).

Remark 15.2. According to Definition 15.2 it is not difficult to note that for A € [R,A * 0, f(x) e 9JT1 (L) if and only if x (A " ~ 1)/2 f(x A ) c,y

JJT* . (L). Consequently, 3Jf* c/ IA|,y | A|,y

a+(j} 1)/2

if we set h(x) = x

A

~

W

f(x )g(x )

€ in

(15.20), where A,w e D OR, Au * 0, then we obtain more general statement than Theorem 15.3. In this case the corresponding pair (c , ,yy )) is is defined defined 3 3

as follows ), [ (c /|A|,y ), l 1

(c 3,y (c ,y3)) == i 3* °3

I

1

if c /|A| < c /|w|,

l

1

(c /|u|,y 2), (c2/|.|,y ), 2

22

2

I (c /|A|,min (y ,y ,y +y )) )) ,, v

§16.

1

22

if c /|w| < c /|A|, ifc/|w|
1122

1

iiff Ci c /|A| /|A| == c 2 //||ww| |. . 2

1

X One-dimensional H-transform i n t h e s p a c e s ffl JJT(L) (L) and 3Jl" 3H a 1( L (L) )

c, y

In §15 (15.3)

and

we have its

introduced

inverse

formula

the one-dimensional (15.5).

In

the

Mellin

present

transform

section

we

establish one important property for this transform and its applications to generalize the H-transform

(15.1) with the Fox's H-function in its

kernel.

129

The results of the present paragraph are very useful for the further studies of the convolution theory considered in Chapter IV. The following theorem is true.

Let in the

Theorem 16.1

strip

s € C, 7) < Reis)

(16.1) a function

F(s) be analytic

Let also the function

function

< 77 ,

and F(s) = 0(|s|~

f(x) be defined

), A > 0, for

by the inverse

|lm(s)|—> oo.

Mel 1 in transform

of

i.e.,

F(s),

(16.2)

f(x) =

l

2ni

F(s)x ds,

Re(s)=v where

77 < v < 77 . 1

Then exists

2

the

and it

Mellin is

transform

equal

(15.3) f (s) of

to F(s) for

all

complex

function numbers

f(x) (16.2) s, belonging

to

77—1

strip

(16.1). Moreover,

f(x)x

e L(0,+oo) for

all

77, such

that

77 < 77


since

the

function

F(s) in

e L(0,+oo) for ri < TJ < TJ . In strip

(16.1)

is

analytic

and

F(s) = 0( I s I " 1 _ ),A > 0, for |lm(s)|—» 00, then equality (16.2) holds for all v,

such that 77 < v < T) . Hence

(16.3)

where x > 0 , 7 7

1

2

|f(x)| < x"y - i ^

|F(s)ds| =

M x

V t

< v < 77 . Further from 77 < 77 < 77 it follows that there

exists sufficiently small c > 0, such that 77 < 77-e < 77+e < 77 . Applying inequality (16.3) for v = 77+e, x > 1 and v = 77-c, 0 < x < 1 we obtain 130

If (x) I < M x _ 7 ) + C ,

f o r 0 < x < 1,

I f ( x ) | < M x" 17 G ,

forx > 1

T7— 1

e L(0,+oo).

Consequently, f(x)x

Now we must prove that if TJ < Reis)

< 7) , then f (s) = 9JHf(x);s} = F(x),

i.e.,

Re(s) +ioo (16.4)

F(s)

X

dX

r

TT

R e (s) + i oo

N

F(T)X

-xdx , =

2ni Re(s)-ioo

lim

x

dx =—7

F(T)X

2ni

N—>+oo

i/N

"dx.

Ke(s)-ioo

The repeated integral in the right part of (16.4) absolutely converges and we can change the order of integration, and after evaluating of the inside integral we have

Re (s) + i oo (16.5)

1 2ni

I(s,N)

Re (s) + i oo

F(T)X

7T1

Re (s) - i oo

Further, using (1937)

and

the

technique

replacing

the

F(x)sh[(s-x)logN]

1

dx

dx.

Re (s) - i oo

of Fourier

variable

in

integrals the

by

E.G.Titchmarch

integral

(16.5)

we

get

the corresponding case of Fourier representation for the function F(s) € L(Ke(s)-ioo, fie(s)+ioo) and moreover it is analytic in the strip which contains the contour of integration. Thus the Mellin transform of f(x) exists and it is equal to F(s) = lim

I(s,N). Theorem 16.1 is proved.

■

Now we consider the classical H-transform and its generalization in the

spaces

JJf^L)

and ffl"1 (L).

In

1970

C.C.Gupta

and

P.K.Mittal

introduced and studied an integral transform whose kernel is the Fox's H-function

defined

by

(1.5).

This

important

defined and represented in the following manner

131

integral

transform

is

(16.6)

(a ,a ) n ' p p H xu p.q (0 ,b ) T T m,

(Hf)(x)

q

f(u)du.

q

Various properties and inversion formulas of the H-function were

also

studied

by

K.C.Gupta

and

P.K.Mittal

(1970), R.G.Buschman and H.M.Srivastava kernel of transform integral

(1.5),

(1970,1971),

transform R.Singh

(1975). Here we note that the

(16.6) is Fox's H-function defined by the contour

which

converges

only

under

some

conditions

of

the

parameters m,n,p,q and (a ,a ), (0 ,b ). These conditions were obtained p

p

q

q

by A.L.Dixon and W.L.Ferrar (1936) and they will been also shown below. Vu Kim Tuan (1986c) generalized the H-transform (16.6) in some space L . In this section we will consider a modification of Tuan's generalization in the spaces 9ft (L) and 3ft (L). First note that if f(u) is replaced by c, y l/uf(l/u) then transform (16.6) can be written in the following form

(16.7)

(a ,a ) } p

m,n p.q

(Hf)(x) =

P

q

where $(s) is defined by

f (u)—

O ,b )

u

=

^-^

2TTI

q

$(s)f (s)x ds,

Re(s)=y

(16.9) below and f (s) is Mellin transform

(15.3) of f(x). Consequently, we introduce Definition 16.1. The H-transform of function f(x),

x > 0, is called

the next integral (16.8)

f(a,a) i,p (Hf)(x) = Hm,n p.q (0,b)

Kf(u)](x)--sr

$(s)f (s)x ds,

l,q

where
= 1/2}

, O^m^q,

m

"TT (16.9)

*(s)

=

j. = i

O ^ n ^ p ,

n

n

r ( 0 +b s) F T T ( l - a -a s ^j

-j

j=i

p

j

j

q

FT r(a +a s) "TT r d - 0 -b s)

j=n+i

j=m+i

132

and the function f (s), generally speaking, is such that f(x) is its inverse Mellin transform (15.5). The parameters of vectors (a,a)

= (a ,a ),

1,

(0,b)

p

p

= (|3 ,b ), . . . , (j3 ,b ), 1

1 , q

are such that a ,...,a ,b ,...,b 1

(a ,a ),

1 1

P

p i

1

q

q

are positive and a ,...,a ,8 ,...,6 q

^

l'

p " l'

'' q

are complex. Moreover, suppose that for them the next conditions take place /te(0.) + -^- > 0, J

j = 1,...,m;

2

1 - Re(a.)-^j

> 0,

2

j = 1,

,n;

(16.10) Re(a

j

) + -i

> 0,

2

1 - /te(|3.)- —

j = n+1,...,p;

> 0,

j = m+l,...q.

Definition 16.2. The ordered pair (K,/I), where m (16.11)

n

£b j=i

(16.12)

p

+ Ea -

E

j=i

P

q

j=i

j=i

q

a - £ b

j=n+i

j=m+i

q

p-q

I a, - E

li = Re

j=i

is called the index of the H-transform (16.8).

The following

theorem gives a relation of the transforms

(16.6),

(16.7) and (16.8).

Theorem 16.2. The H-transform and only

if

the

next

condition

(16.8) exists holds

valid 133

on the

space

3JT (L) if

2sgn(»c) + sgn(jn) ^ 0.

(16.13)

In

this

takes

case

(Hf)(x) e !JJ1 (L). If,

the

next

inequality

2sgn(/c) + sgn(ju-l) > 0,

(16.14)

then

moreover,

place

the

following

representation

of

the

H-transform

(16.8) can

be

obtained

H(*)f(u)^ , u u

(Hf)(x) =

(16.15)

where H(x) is Fox's

Proof.

With

H-function

(1.5).

the aid of the asymptotic

estimation

(4.2) of the

gamma-function and Definition 16.2 for the kernel $(s) defined by (16.9) we get the next relation

| Im(s) | ~M1 ,

<Ms) = o{exp[-nK\lm(s)\]

[16.16)

+co, Re is)

= — , 2

Hence from condition (16.13) it follows that sup|$(s)| < C < +oo. Since f(x) € 3H (L), then from Definition 13.1 we get $(s)f (s) e L(cr), that is why (Hf)(x) exists and belongs toffl(L).

Now if inequality (16.14) holds valid, then from (16.16) it follows that $(s) e L(cr). Hence the Fox's H-function (1.5) exists provided that condition

(16.10) holds. Further

applying

Theorem

16.1 about

Mellin

transform we get that if inequality (16.4) is valid, then

f ,(a ,a ) 1 3H

j H(x); s|

p

p>q

p

(0 ,b ) q

q

134

x

dx = $(s),

for s e cr.

Therefore we can apply Mellin-Parseval equality (15.4) to (16.8) and we obtain 16.2.

representation

(16.15).

This

completes

the proof

of Theorem

■

The following

theorem

gives

an operation

of the H-transform in

m'1 (L). Theorem 16.3. The H-transform (16.12) exists

1

on the space

(16.17)

HTf

(16.8) wihh

the index

(L) if and only

(jc/fi) (16.11)-

if

2sgn(K+c) + sgn(/i+y) > 0.

Furhhermore

theee

is

isomorphism

by H-transform

betwenn

3J11"1 * (L) and c,y

HIT 1

((L). L),

c+K,2C +M C + K , y*\x

Proof. sV

c | ! K

s

The function f * ( s ) « LC<0.

f(x) belongs For f u n c t i o n

JJ1 1 (L) to JTf1

if and only if

c,y

* ( s ) (16.9)

we h a v e

t h e next

eestimation stimation

< 00, oo, s s 6 6 (cr
TT

y

n 7Tc| II m m ss

Hence s e e Hence s L(
° \

I| r* * t,

\*

. t, T

\. •. r>_

J,

i.

-- jj r j* ^r f \ Ml

M 7T(c+K) 7 T ( c + K ) || IInm ll ^^++M s s|| **

f ( (.sj and only o n l y if i f $(s)|s| $(sj|s| f .sj € e Llcrjif Llcr)if and

e e

..

f ((s) sj e e f

On the other hand ||(Hf)(x)|

m *

=

(D J

[|$(s)s^e7r(c+K)|lms|f#(s)ds|

C f[ | Ss ^Ve c7 r| cl |"l ml fs |*f *( (s s) )ddss | t* C 2

and 135

=C C | f ( xx))| | = 22

1 V ) V 1 (L ,D

|s3re7lc|lms|f*(s)ds|

|f(x)||

|$(s)s^e7l(c+K)|lms|f*(s)ds|

E «(Hf)(x)|| 1 UR

(L)

C + K , y+ii

Hence the proof of Theorem 16.3 is completed.

Theorem 16.4. The next

H-transform

-1 (a,a) m,n '1'p|| •[g(u)](x) p>q O.b) l.q'J Hp-n,q-m

(16.18)

q.p

f(|3fb)

1 2711 *(s)

the

inversion

1,m

I (a,a) v

is

,(0,b) m + l,q

for

>[g(u)j(x)

,(a,a)

n+l , p

1,n

g (s)x ds

H-transform

(16.8) in

the

space

!U1

(L). Here

c + K . ^ + fi

(a,a)

and (B,b)

1,P

are defined

and (
. (1 < k ^ j) means the vector

( Proof. belongs *

in accordance

p

k

k> j

In fact, from Theorem

to the space

JH

H-transform

(16.8J

*

and its parameters

of well

regulated

components

z ), . . . , (
j

j

16.3 it follows

(L) and

that

according

g ( x ) = (Hf)(x)

to (16.8)

w e get

-i

g (s) = $(s)f (s). Further [<Ms)] product

with

l,q

have

is also the ratio of gamma-functions the arrangement

which

is defined by

(16.18). Hence the following H-transform (16.18) h a s the index pair (-K,-/I) and (Hg)(x) belongs to !U1 ( L ) . Further substituting the product c, if

136

$(s)f (s) in (16.18) with the aid of the definition of the space 9JT (L) we get the statement of Theorem 16.4.

Theorems H-transforms

16.3 and

16.4 show

■

that

a

is also a new H-transform,

composition

and moreover,

of

any two

the identical

transform (Ef)(x) is included in this class H-transforms.

>["f(u)l(x )

0,0

Theorem 16.5. In accordance

with

= (EfMx) = f(x).

Definitions

16.1 and 16.2 let

f (a',a'! I,P

(H'fMx) = H m , , n , P >q

(16.19)

»|"f(u)j(x)

(0\b') l,q

be some U-transform

with

the

index

(K'./LI'). Then the

composition

of

two

H-transforms

(H'o H)(f)(x) =

(16.20)

exists

on the

following

space

inequality

(16.21)

!JH

c, if

(L) if

is also

H'o (Hf)

and only

if

(x

together

with

(16.17) the

true

2sgnU+K'+c) + sgnCju+fi'+y) ^ 0.

In this

case composition

(16.20) is also

[H>O ( H f ) i ( x ) = H m + m !' n + n ; L J P + P >q + q

(oc,a)

a new

H-transform

, (a' , a' ) , ,

1 ,n

(£,b)

n

, (/3\b')

1 ,m

,, 1,m

(16.22) (ex,a)

, (a' ,a' ) ,

n+l , p m+l , q

and it

maps 3TI

(L) isomorphically

ff(u)l(x)

n +1 , p

, (/3,,b>) ,

0,b)

,

m +1 , q

into

c,J

3J1 c+K + K

137

j

'

, >y

, (L) + ll + l±

Proof. Indeed, here condition (16.17) provides the existence of the H-transform

(16.8) on

isomorphic to UJf1

the

space JJf1 (L). In

(L) by H-transform

JJf1 (L) is

this case

(16.8). Further H'-transform

c + K . ^ + jLl

1 (16.19) (16.19) exists exists on on the the JJf JJff

c+K, "+ fl

is valid. Finally,

(16.22)

(L) if if and and only only if if that that condition condition (16.21) (16.21) (L)

is accounted

by

the

fact

that

the kernel

of

composition (16.20) will be equal to $(s)$'(s), where $(s) and $'(s) arr the kernels of H-transforms 16.5 is proved. The

(16.8) and (16.19), respectively. Theorem

■

following

theorem

gives

composition

structure

for

the

H-transform (16.8). Theorem 16.6. According

to Definionsns

16.1 and 16.2, let

(Hf)(x) k

be

some

H-transforms

k = l,2,...,r. Let kp =PD2,<
C H f H x ) as

dollows

with

also tae

the

the

index

kernel

nrodnck

of

$(s) (16.9) of the*

respectevely

(K ,JLI )

kpmp

the

for

H-transform

1
thp the*

(16.8)

H—ti~An
CHfHx) as dollows (16.23) Then,

$(s) = if

the following

r r 1 T T k= =l l' k

conditions

$ k(s). (s). k

are

sasasfied

J J

(16.24)

for

all

Jj

2sgn(c + V K ) + sgn(? + V ua ) > 0 kti k kti k j = 1,2,...,r, then

can be represented

in

the

JJf1 (L) the

space

by the composition

H-transform

of H -transforms

as

(16.8)

follows

k

(16.25) This

(Hff(x) ) (Ho H r r

theorem

immediately

r-1 r-1

follows

0...0Hf)(x). 1 1

from

Theorem

sufficiently remark that the composition (Ho H ^ ^

l

1

16.5.

Here

we

0...<>Hf)(x), 1 ^ 1 ^ r, l-i 1

-

1

1l

exists on the space JJf1 (L)if and only if (16.24) holds valid for all c,7 exists on the space JJf1 (L)if and only if (16.24) holds valid for all j =

1.....1.

j = 1.....1.

138 138

Now we consider four most simple particular cases of the H-transform (16.8), when p+q = 1. Let a,/3 € C, a, b e (R such that

(16.26)

Re(|3) +

— >

1 - Reia)

0,

Then we have four following

[i6 27)

H

o;!(o;b>Hftu)](x)

(16.28)

H

i,o[

-

(16.29,

H

-

=

^7

J ° [ f ^ ) j ( x ) = 2ST

is

S : ? ( ( a ; a ) ) ° [ f ( u ) ] ( x ) = 2si

not

difficult

to

obtain

inversions of the H-transform

(16.31)

H-transforms

H;;;^).^)]^.^

ti6 3o)

It

-

H o.o(, P; »)

1

,1,0 0,1

H°'°f " 1 0,l[(ct,a)J

,0,1 1.0

The following theorem is true.

139

f (s)x

ds

T ( l - a - a s ) f (s)x

I

f (s)x

ds

ds

T(l-a-as)

(13.b)

((oca)}

ds

TO+bs)

that

and

(16.32)

r(|3+bs)f (s)x

(16.27) and

,

- -> 0.

(16.28)

and

(16.30)

(16.29), respectively,

are

the

i.e.,

16.7. Let

Theorem (16.8) with H-transform

the of

index

condition

(16.17) hold.

(K,JU) can

types

(16.27) and

be represented

Then

the

H-transform

by compos it ion

(16.29) as

of

p+q

follows

((a,a)

; b ) H.[ f
p>q

HUO,1((0m+1 ,b m+1 )}

R0,lf(/3

H°>°(

H°>°( 1,0 v (a ,a )

i,o[

(16.33)

_

l,ol(a v

J

,a

n+l

p

n+1

„0,l((a ,a )} oH

i,o[

*_

r

J °---°

u 0,lf(a

,a ))

H

_n

n

i,o(

q

y

>|"f(u)ji (0 ,b )•■■■• H i ; ! o ,b )J-If(u)|(x).

<01 v

1

1

y

v

Proof. Set $ (s),...,$ 1

f 1

,b )

m

m

y

(s) as follows p+q

$ (s)

= r(0 +b s ) ,

j

j

j

j = 1,...,m;

$

(s) = T(l-a -a s) ,

j = l,...,n;

$

(s) =

j = n+l,

m+j

j

j

(16.34) m+j

j

$

j

j = m+1,...,q.

.(s)

P+J

r(l-|3 -b s) j

j

Hence the kernel $(s) of the H-transform product

p;

T(a +a s)

(16.8) is equal

to the

(16.23) of $ (s), k = l,...,p+q. In accordance with (16.11) the

indices K

k

of the kernels $ (s), defined byJ (16.34), can be k

follows

140

written

as

K

= b /2,

j = 1,...,m;

=a/2,

K m+j

= -a /2,

K m+j

j = n+1

p;

J

j

P +

j

n;

J

= -b /2,

K

'*"

j = l

j

j = m+1, . . . , q. J

j

»

» H

Further, it is evident that the index (K,JLI) of the H-transform (16.8) can be as follows (K,M)

Since all a , b j

=

are real positive numbers, then we have K j

> 0 for

k

k = l,...,m+n and K < 0 for k = m+n+1, . . . ,p+q.

Hence, we can write K in the

next form m+n

Ei^i-T E hi-

l*r j=l

j=l

"

j=m+n+l

Therefore, it follows from (16.17) that 2sgn(c+K) + sgn(r+^) m+ n

= 2sgn

-+EiKji j=l

Now

the last

p+q

T

- E i K ji

"

j=m+n+l

inequality

allows us to conclude

(16.24) hold for all j = l,...,p+q composition proved.

We

representation

0.

+ sgn

(16.33)

that

conditions

(here r = p+q). It means that the is valid.

Theorem

16.7 is thus

■

note

that

from

(16.26)

by

Slater's

residue

theorem

(see

O.I.Marichev (1983)) we get

(16.35)

Vl[X

(0,b)J

" Zni

<«•»> W l - s i

>M-

r(/3+bs)x' s ds = b ^ ' " e x p

_,.

.

T(l-a-as)x

141

-s,

ds = a

-1

(a-l)/a

x

exp

[,-<■]

Consequently, Theorem

16.2 (see also

(16.15)) allows us to represent

(16.27) and (16.29) as follows

<"■*" HJ:;U; b ,HH ...».

(x)

,/3/b

exp

-rK)

H^-) ] f ( <

■£;(<■■•>).[«.,]«„ - f.-(i)'""-„[.(i)-'-]„ 0) s

In the case b = 1, /3 = 0 and a = l,a = 1 we obtain modified Laplace transforms which are considered in detail in § 1 7 — § 1 8 .

§17. The G-transform and its special cases

The present section is devoted to the particular case of the H-transform (16.8) when all coefficients a and b are equal to unity. j

j

Naturally, this transform is called G-transform and in accordance with Definitions 16.1 and 16.2 we introduce the following definitions.

Definition 17.1. The G-transform of function f(x), x > 0, is called the next integral

f (a) (17.1)

(Gf)(x) = G m , n p>q

Tf(u)l(x)

where cr = {s e C, Re(s) = 1/2 >, 0 ^ m < p ,

0
77 ro.+s) 77 ra-a.-s) (17.2)

*(s) =

j=i

j=i

p

77 j=n+i

q

r(a.+s) 7 7 rd-p.-s) J

*(s)f (s)x ds, 2712

(0)1,q

j=m+i

142

J

and f (s) such that f(x) is its inverse Mellin transform (15.5). Here we also assume that — -Reioc

/te(/3 ) + — > 0, j = l,...,m; ' j

(17.3)

,

2

Re{ocJ+— J

J

2

) > 0, j = l,...,n; j

> 0, j = n+l,...,p; —-fie(/3j > 0, j = m+l,...q.

2

2

J

Definition 17.2. The ordered pair (c ,^ ), where P

p+q 2 '

(17.4)

q

Re J=l

J=l

is called the index of the G-transform (17.1).

From Theorem 16.3 it follows immediately that

Theorem 17.1. The G-transform on the space Jit

(L) if

and only

(17.1) with

the

index

(c ,f

)

exists

if

C, Q

2sgn(c+c ) + sgn(^+^ ) > 0.

(17.5)

m

In

this

#

#(D.

c+c

case

the

G-transform

maps 5J1

(L) isomorphically

into

,v+y

Now applying Theorem 16.4 we see that the inversion of G-transform (17.1) is also G-transform and it can be written as follows -1 ■

G m,n

p.q

((a)

I,P

V

•(g(u)j(x

(£) 1 ,Jq ^

y J

(17.6)

= Gp-n,q-m

q>p

m+1

(a)

'q

1 > m

, (a)

n+l , p

2ni

hHs)

og( U )

IL

1,n '

g (s)x ds.

143

(X)

J

The following theorem represents our G-transform defined by contour integral (17.1) in the traditional real form.

Theorem 17.2. Let (17.7) then the

the real

2sgn(c ) + sgn(? -1 ) > 0, G-transform form

as

(17.1) with

the

r

(17.8)

index

(c ,y ) can be represented

in

follows

(Gf)(x) =

(a) }

X

G m,n

p

du f(u)

p.q u

where

Gm,n "p.q

(a) 1 p

is

Meijer'

s G-fund

ion

(1.3).

(0)

Here the existence of the G-transform (17.1) is guaranteed by (17.7) and inequality (15.7) in Definition 15.2 of IJJf1 (L) (see also (17.5)). Theorem 17.2 shows that inequality

(17.7) provides the convergence of

the Mellin type integral (1.3) which is used for definition of Meijer's G-function at least for the contour L, coincided with the line
Remark 17.1. If inequality (17.7) is replaced by 4sgn(c ) + 2sgn(3r ) + sgn|p-q| > 0 then the statement in Theorem 17.2 is also true (see Vu Kim Tuan, 0.1. Marichev and S.B.Yakubovich (1986)).

Note that classical G-transform (17.8) in the next equivalent form r

* (17.9)

(Gf)(x)

_m,n G xu p.q

,

was also

studied

Kesarwani

(1962;

by

several

1963

(a) ) P

f(u)du

q'

mathematicians

a,b;

1965

V.K.Kapoor and S.Masood (1968)). 144

a,b;

(see C.Fox

1971),

(1961),

V.K.Kapoor

R.N.

(1968),

Further, since the Meijer's G-function

(1.3) is rather a general

function, then from (17.8) any its particular case defines the corres­ ponding transform, which may be known or unknown. Here we give the table of those transforms which are required to study the convolution theory in Chapter IV. For convenience we introduce the following notations of transforms

(see also Brychkov Yu.A., Glaeske H.J.

and Marichev 0.1.

(1983))

(17.10)

k( x)joff(u)l

(17.11)

x \ ( x ) U f ( u ) j = r ^ j a k ( 5 ) f (U)«H = x a | k ( x ) U u - a f (u)

k(*)f(u)*H u u

2ni

k (s)f

(s)x ds,

= (xak(x)x"a)(f(x)), - 1

1

(k(x)} .[f(u)] =-±j

(17.12)

*

k (s)

-s

f (s)x ds,

- 1

-1

(17.13) (x\(x)} .[f(u)] = ^ i J

Below

we

give

table 17.1

1

f (s)x ds = (x k (s+a)

of

definitions

of

a

{k(x)lx-a )(f(x)).

important

simplest

G-transforms in forms (17.10)-(17.13), where the function *(s) (k (s)) is defined by

(17.2) such that p+q ^ 2. These transforms are special

modifications of known classical integral transforms and their inver­ sions

(Laplace

transform,

Hankel

transform,

Stieltjes

transform,

Riemann-Liouville integro-differential operators, Meijer transform). The more general particular cases of G-transform, which will be introduced in Chapter IV, can easily be obtained from (17.1), (17.10)-(17.13) by the using the table of Mellin

transforms and representations of the

kernels

G-function

k(x)

through

Meijer's

(1989)). 145

(see

A.P.Prudnikov

et al.

TABLE 17.1 Simplest integral G-transforms and their inversions

Modified Laplace transforms

00

■ <°[ 2

G°'*

oI

-(x/u)_, .du f(u)—

.ff(u)l(x) = { « " * } • [f(u)] " < A / H x > = f 0 00

1] .[f(u)](x) = l e " 1 / X H f ( u ) ] = (A-f)(x)

=

f

-<x/u)~*f (.du u)—

0 0

3

' -(x/u,*1 -«fl ,du {xVx±1}o[f(u)] = ( x < V a ) ( f ( x » = x * e u flu)— c) -1

4

1 ~" r 1,0 G

o,i

I°

»[f(u)](x)

= { e - * } »[f(u)] = (A;'f)(x)

= _i_

[_JL_ f*(s)x-sds

J r(s)

5

f V x _ 1 l .[f(u)l = (xV^HfCx)) - - J L

1 f*( )x_sds J. 1 Os J Jv U O T(±(s+a))

——^^^———^^^—

I

146

Modified Riemann-Liouville integro-differential operators and their inversions

6

GG l° '*l jf[

a a

X "1)++ l| °o [[ ff (( uu) )]l((xx )) == J| ((X "1)

|orf(u)l == I^(xI^(x-aaf(x)) f(x)) V[f(u)]

x

r (x-u) a_i -«_ ,_,

= [ ' x - " ) 0 6 " 1 u-«f(u)du, = J T(a) u f(u)du, Jo T(a) o x

Re(«) Re{a) > 0

x

1 nn - «f(u)du, , . ,A,A = d f ( xx--uu))""**" - u-«,, = dxnJ Ha+n) u f(u)du, dxnJ r(a+n) oo

Im(oc) 1,2,3,.. 7/n(a) * *0,0,n n= =1,2,3, ..

-n < Reioc) Reia) < <-n+1 -n+1 oror ReRe {a){a) = =-n, -n, 7

Gl jf ':lPL f G!'?[

aa

|o[f(u)l(x) = j

1, [[ 0 J L J

X )) ++ (( 11 X

"

([

lo[f(u)l = Ia(x"af(x))

r(a) f(a) J L J

00

=

l u XJ

u'af(u)du,

r(a) J r(a)

/te(a) 00 Reioc)> >

xX

00 00

ff = =

dd ^[(u-x) ^[(u-x)

a+n_1 a+n_1

l" r ^^ JJ JJ rr(a+n) (a+n) x

- n < Reioc)

< -n+1

- n < Reioc)

< -n+1

or or

-a-, ,. , , -a-, (UMU

" f(u)du' ' ' u

x Reioc)

= - n , I/n(a)

* 0, n = 1 , 2 , 3 , . .

Re(oc)

= - n , I/n(a)

* 0, n = 1 , 2 , 3 , . .

-l

8 8

|

(±1+X X ))

+*

|1 . [ f (l u ) ] = ((l^)"(x-af(x)) l ^ ) " ( x - a f ( x ) ) = i ; a ( x "- a f ( x ) )

147

Modified Hankel transforms and their inversions

9

1,0 0,2

|~f(u)l(x)

G

= j j ^ U v S o j o |~f(u)l

iV2,-iV2

J V (2/^7U)f(u) —u

10

0,1 G, 2,0

1-IV2,1+IV2

M ( x ) = -KbHHH J (2/~TI^Of(u) — V

u

-1 11

a,o J

0,2

Tf(u)

J {2Vx)\

(x)

o f(u)

V

iV2,-i>/2

'-,J.tHM'H -1 12

.0,1 J 2,0

fl-iV2,l+iV2)

■Htx) = Wi)} °M xJ ( 2 \ / x ) U | f (u)l

148

Modified Sin- and Cos- transforms and their inversions

13

G

1,0 0,2

Tf(u)j(x) = j—sin(2v^)|o["f(u)l 1/2,0

_1

14

G

^ 1,0 ( |"f(u)l(x) = |—cos(2V^)|orf(u)l 0,2 0,1/2

_1

15

0,1 G. 2,0

0,1 G.2,0

cos(2i/x/u)f (u

du

f 1/2,1 o|"f(u)l(x) = j—sin(2x'1/2)|o["f(u)l

_1

16

,du sin(2yx/u)f(u)— u

sin(2/u/x)f(u)du

( 1/2,1/2 »["f(u)j(x) = j—cos(2x"1/2)|orf(u)l

J. cos(2^/u7x)f (u)du

149

r

r

17

(/

|G^O G°;J

-1 -i \\ ~\

--1l

Mof [(fu( u) )] ]((xx)) = | - i s i n ( 2 vV^x))}|

=

o[f(u)] .[f(u)]

M / ^ x -x1-/ 12/s2lsni n( 2( 2xx- -l /12/ 2) )l loor[ff ( u ) l U? JL ViF J L J J

-l -1 18 IB

l l o .frff((uu))ll (( x ) -=((^- ci coos s( (22VVxx))\\11..rf ff (( u ) l [V G ° M A ~ 11 L J 1J 1 |[ 22 '' 00 (l oo ,, ii // 22jjJ| L J l[^ /i L 1 J ( 1 i -1/2 -i/2 r,., J -1/2.1 r\el, ,1 o -1/2,1 <— x JM ) ) = ^— cos(2x H ° f (ful u

-1

19

k'jf 1/2,1114f(u)](x) = {-sin(2x"i/24 °[f(u)] [ G ° ' j [ 1 / 2 , 1 | | oTfCujjCx) = U s i n ( 2 x - 1 / 2 ) |

o[f(u)]

/2 = jp- A x sin(2^)}o[f(u)l = x11/2 sin(2^)}.[f(u)l

20

20

-1 r 1 r 1/2,1/2 n

-1 1 f n ,( 1/2,1/2 l l r i /• , i"1 r i 1/2 n r G°;J . [ f ( u ) ] ( x ) = | - i c o s ( 2 x -1 / 2 ) | . [ f ( u ) ] G°;J o [ f ( u ) j ( x ) = j - i c o s ( 2 x - ) | o[f(u)j

n

/-

r r i

= _ ! x 1 / 2 cos(2yx)}o[f(u)l cos(2v^)|orf(u)l = j|_L

150

Modified Stieltjes transform and its inversion

21

G

1,1 { 1-P 1 >|~f(u)l(x) = jr(p)(i+x)~pjorf(u)l 0 1,1

uPf(u) du

Tip)

(x+ur

,1,1 '1,1

22

1-p 0

>[f(u)l(x) = | r ( p ) ( l + x ) " P |

o|"f(U)l

1

1

2ni

r(s)r(p-s)

f (s)x ds

Modified Meijer transforms and their inversions

23

G

2,0 0,2

>|~f(u)~|(x) = J2K y (2v^)|orf( u )l y/2,-i>/2

= 2

24

0,2 G. 2,0

l-iV2,l+y/2l

K (2/^7U)f(u) — V u

M(x)-Kkr)}= 2

f(u)

Kv (2\Hi7x)f(u) du

151

25

.2,0 J 0,2

>|"f(u)](x) = J2K^(2V^)|

1 2ni

26

.0,2 J 2,0

o|"f(u)l

v/2,-v/2

f l - v / 2 , l+v/2'

r(s+iV2)r(s-i>/2)

f (s)x

ds

■H(x)-htr)} °M i 27T1

1

r(-s+v/2)r(-s-v/2)

Integral transforms with Kummer

f (s)x

F (z) and 1 1

Whittaker

27

W

(z)

functions

G 1, 1 [ 0 ^ _ b J o [ f ( u , ] ( x , = { r [ ^ ] i F i ( a ; b ; - x , } « [ r ( u ) ] 1,2 00

= r [ M [ F (a;b;- 5) f(u )du [_ b J i i

28

1,2 G. 2,1

u

u

fl/2+v,l/2-i/l ff(u)l(x)

= jr[l/2-p-y,l/2-p+Je(2x)

-p 00

W p ^(i)Wf( u )j = rJl/2-p^,l/2-pJ ie (2x/u)

152

.,

W p,v

,U,._,

.du

(-)f(u)— x u

ds

§18. Composition structure of the H- and G-transforms

As it was introduced

in §17, the modified Laplace transforms with

power multipliers and their inversions are defined as follows

[18.1)

(xaA x"a)(f(x)) = +

^

f(s+a)f (s)x ds

2ni

e'x/u u~af ( u ) ^ , u

(18.2)

ReU)

> -1/2;

(x"aA xa)(f(x)) = - U r(a-s)f (s)x~ ds Zni

-u/x a_, sdu e uf(u) — , u

(18.3)

, a -l -a., _r . . 1 1 f (s)x (X A X ) (f (X) ) = ^—r + 2ni f(s+a)

(18.4)

(X

A

X ) (f(X) ) = ^ r

2ni

They

are

simplest

particular

, . Reioc) >

ds, ReU)

/0

1/2;

> -1/2;

1 f (s)x~ ds, Re(a) > 1/2. f(a-s)

cases

p+q = 1.

153

of

the G-transform

(17.1),

when

According

Definition

17.2,

the

corresponding

(18. 1 )-(18.4) are equal to (1/2, -Reoc) ,

transforms Reu)

to

indices

(1/2, 1-Rea),

of

(-1/2,

and (-1/2,-1+Rea), respectively. Hence they map the space 3J1

(L)

c , if

i s o m o r p h i c a l l y i n t o IJJf1

( L ) , JJT 1

c +l/2,3T-Rea

JH

fflfl"1

(L),

c +1/2, y + l - R e a

(L),

c-l/2,2r+Re(X

( L ) , respectively. c-i/2,r-i+Rea In the present

G-transform

section we find composition

representations

by these simple transforms. Here we also give

of the

composition

structure for all known convolution type transforms which are mentioned in §17. In accordance with (18.1)-(18.4) we index all operators of the form

x

13. - £ . J A x J, j = 1,

i-a

J

. . . , p and x

J

x A

x

J

a -l a_ -a A x J , j = l , . . . , n ; x J A _ 1 x J , j = n+1,

,j = m+1,. .. ,q in a single sequence A ,..., A 1

According

to

Definition

17.2 denote

by

# *

(c ,3- ) k

transform A

for k = 1,

p+q

the

index

of

k

p+q respectively. After some calculations, we

k

get f 1

(c ,7 (c

j

A

,y

m+j

(18.5)

(c

m+j

(c* p+j

From

) = (l/2,-Re(/3 )]

j

fy

for j = 1,...,m;

j

m+j

m+j

,/

) = (l/2,Re(a ))

for j = 1,...,n;

) = (-l/2,/te(a ))

for j = n+1,. . . ,p;

) = (-l/2,-/te(|3 ))

for j = m+l,...,q.

j

j

p+j

j

Theorem 16.6 about

composition

representations

of

the H-

transform it is not difficult to obtain the following result.

Theorem space

1

3JT

permutation

18.1. Suppose

that

(L), i.e., conditions

{j , . . . , j 1

} of

the

G-transform

(17.3) and

(17.5) hold.

the sequence

p+q

conditions 154

(17.1) exists Then

in for

{l,...,p+q> satisfying

the any

the

(18.6)

2sgn c + V c.

the following the

space

+ sgn Ur + V" y.

composition

representation

£ 0

for r = 1, . . . ,p+q,

of the G-transform

is valid

in

!ffi (L): c,y

(18.7)

Remark

„m,n J p,q

18.1.

(

(a) 1,P

o|"f(u)l(x) = [A O ...O A f)(x) L J I Jp+q \ J

l,q

Here from

(18.5) it follows

that

c

= 1/2

> 0 for

j

j = l,...,m+n. Hence composition representation (18.7) is valid for the permutation {j ,. j > = {l,...,p+q> (see also (16.33) in Theorem p+q

16.7).

Remark

18.2.

By

combining

the

operators

A

.,A

,

with

preservation of their order, into various groups, we can obtain other composition representations of the G-transform, having, in particular, Riemann-Liouville

fractional

integro-differential

operators

and

the

Hankel, Stieltjes, Meijer transforms, etc.

Example 18.1. Now we consider some composition representations of G-transform with Gauss hypergeometric function (1.1) in kernel

(18.8)

F (a,b;c;-x]

2 l

Tf(u)

F (a,b;c;- - ) f ( u ) —

2 1

155

u

u

According to Table 17.1 of the definitions of simplest operators and the following Mel1 in correspondence between the kernel

F (a,b;c;-x) and its

Mellin transform (15.3) (see formula 8.4.49.13 in A.P.Prudnikov et al. (1989))

IT ( u ^ = F (a,b;c;-x)

2 1

Ml-a,l-bj

T(c)

G 1,2(

r(a)T(b)

Z

'

Zl

(18.9) s, a-s, b-s

r(c) r(a)r(b)

we have the next composition representations

j2Fi(a,b;c;-x)|orf(u)l

r (C)

, . , , -a. a. , (A )o(x A X )o( x A x

)o(x°A

x°)of(u)

r(a)r(b)

= usi (x 1 - 0 ^- x - i )./(i + x)- b i.rf(u)i

(18.10)

T(a)

r(c)

+

I

jx(1-c)/2J

rta)r(b) I

/ L

J

(2^)U{x-(a+b,/2K

c a

"

/ I

(2x-1/2)lo[f(u)].

a b

"

/ L

Below we give the table of composition representations

J

(factoriza­

tions) of simplest operators by modified Laplace operators. Here symbol f(u)

in the left and the right parts of equalities given below is

omitted.

156

TABLE 18.1 Factorizations of Simplest operators

1

/ x V x ) W x PP h ( x ) W f ( u ) ]

2^

a a -a-a x Vi xx"a _ a x i ++ x

= x aa | k ( x ) J o / x // 33 - aa h ( x ) U u - // 33 f ( u ) ]

_ a+a-i 1 i-a-a a-i - i1 1i - a , - ixf x a + a "A_X A x 1 " a ";ao)^ox( x a "AV X x " a ); ix A_X ;o^x A X ;

3

x - 4 - = (x _ 1 A x ) o ( A _ 1 )

3

x - 4 - = (x _ 1 A x ) o ( A _ 1 )

{j^(2^)l = ( x ^ A y ^ ^ o C x ^ - ^ V ^ 1 ^ 2 )

4

5

fT , „ - 1 / 2 . ) |Jv(2x )| f,

,„

-1/2,1

, -V/2A V/2, , 1+1//2 -1 = (x Ax )o(x A+ x , -IV2.

V/2,

, 1+IV2.-1

-\-V/2. ) -1-IV2,

-l

5 6

| -j M = (x j y2(x2 ^ ) | ) | = ( x - 1 - yA/ 2xA _ x )1 .^( /x2 ) o ( x i ;A/ 2+ Ax- 1 x - y / 2 )) -1

6

{V2^1}

7 7

-l fT , „0 - 1 / 22 ,. 1) f, |V2x >J |j^(2x )|

8

99

= (x~1~lV2A/+lV2>°(xl'/Vx~V/2) y //22A. / /22 . , --IV2 V/2. , l+V - 1 - I1V I V 2 . --11 I V 2. = (x \Ax )o(x A_ >) A xX +x

{( -Ji L- s i n ( 2 vV/ x ) l = ( x 1 / 2 A xx"-11// 22 ))o. ((xx-"11AA" ^1 x )

|/ _ J! _ s i n ( 2 x -- 11 // 22 )) l| = ( x "- 1 / 22 A __ xx11//22))oo((xx A A^' xV"11 )

- 1l

10 10

/| A J _- s i nn((22V^x)) ll \.rz

2 A x )Joo( xt x1 / ^ AV"11 = ( x ^- V *

157

x"1/2)

11

1

—

,0

-1/2,

,

) \

c o s ( 2 x - 1 / 2 ) l = (A

{-i-

.

-1,

= (x A x

, -1/2.-1

)o(x

A

1/2,

x

c o s ( 2 V ^ ) l = (A ) o ( x " 1 / 2 A " 1 x 1 /

/J_

12

13

.

sin(2x

)o(x1/2A-V1

-l

—

14

15

16

17

18

19

cos(2\/x)i

1

,_

cos(2x

f(p)

Tip)

, -1/2.

= (x

-1/2. 1

)}

1/2,

Ax

, 1/2.

= (x

,A"U

)o (A

-1/2,

Ax

)

, -1,

)<>(A_

( l + x ) ~ p i = (A + )o(x" P A_x p

1

(l+x)~Pt

= (A"1)o(x"pA"1xP^

r V/2. «' / *A - 1 V/2, --V/ = ( x Ax )o(x A x

J2 ^ ( 2 x " 1 / 2 ) | = (xy/2A_x" V / 2 ,

20

Kv(2i/x)|

21

2 V2x- 1/2 )j

, "1^/2 A

) o (x

V/2,

Ax

)

, V/2 -1 - V / 2 , , - V / 2 -1 V / 2 , (X A X )o(x A X )

, -V/2 -1 V / 2 ,

(x

158

A x

, V/2 -1

J o (x

A x

-V/2,

)

The next table contains Mellin transforms of the kernels for simplest

operators,

which

give

the composition

the

representations of

G-transforms. All correspondences can be obtained by the Mellin-Parseval equality (see also (17.10))

k*(s)f*(s)x s ds.

(18.11)

L

TABLE 18.2 Mellin transforms of the k e r n e l s for the Simplest

1

kk**((ss)) < <

2

k*(s+a) <

. 11 m

3

kk**((ss++aa))

A 4 4

r1 (<*+<*) is+aj 1 is+aj

operators

>> J| kk (( xx ))l|

> | x aa k ( x ) |

< < >> jxjx0C0Ck(x)}" k(x)}"11 I I

4 < <

J J

»>/*a * x e"x e 1 |- xaA x A + x"a X >| x e | - x AA+x+ X j

> |{xx--aaee--11// XX lj = x"aA_xa

5 5

r(a-s) <

6 6

r(s+a) r ( s + a ) *< Hs+a)

x x *> i1 xX ee / } "" x \♦x

1 , « r(a-s) * * ru-s)

, / - a -i/x\" -i/x\_1X _ - a - li a > \ / ~ " x A" x * \x e /

7

1

{a a -x\

159

_ a -l -a

T(a+s)

Jxa(l-x)^~a M _

r(a-s)

Jx^^x-l)^-1!

a |3-a -(3 -0

8

9

^-

T "~ ^

r(a+s) T(a+s) 10

rnoe - s ) *<

10

11

12 12

113 3

.14. 14

1C 15

/ (a-/S+i)/2. (a-/3+i)/2T

}

i - ^T P-a i-P - a a-i

I++ xX

r, o, / - ^, !l

J

>\* > \x

= x =X

UVx) a*P-ilZVx) f oc+p-i

■»M,iw4**lw ^ 2 ^ 1 — {^"~-^4 r ( 3S / 2 - «g - s ) <<

T(a-s) r(a-s)

r(a-s)

ro/2-a+s) '<

16 16

r(a+s)r(p-s) r(a+s)r(|3-s) <

r(a+s)ro-s)

L -1/2II -l/al)

J

a.p-i[2x

*{*

r(a-s) r ( l / 2 - a + s ) *<

1^

» {{-— ^ -xxaa'-11//22ssiinn((22 v ^ ) l|

>

/ /2, 0-a-i)/2 T

IT^iT'

15

17 17

|

r (P-a; M n

))

/ 1 -a ,„ --1/2,1 1/2,1 * \ ^ X C°S(2X >J7

/ 1 -a+1/2 . r._ -1/2,1 o -1/2,1

*> \^rx \~x

nt s iin( 2x

'J '[

a a p > |r(a | r ( a ++ p)x | 3 ) x a(i+x)" ( l + x ) " a ""/ 3 |

1 <—* ^ _ / r ({r( « a+,„x*u x « ( i + xr«-nx)-n" v

160

+P

+

1

J

(a+/3)/2

r(a+s)r(/3+s) << r(a+s)r((3+s)

> | 2J2x » x(a^)/2K

r(a-s)rO-s) <

> |2x-(a+/3)/2Ka_/3(2^)i

19

r(a-s)rO-s) <

» |2x- ( a + P )/2Ka/3(2^)|

20

1 r(a+s)rO+s)

18 lo

19

I

1

20 21 2i

z <

,

r(a+s)r(0+s) < r(a-s)f(p-s) < na-s)r(e-s) *

22

r

23 23

r(s- P )r(i/2-i;-s)r(i/2+y-s) )r(i/2-v-s)r(i/2+v-s) <-

K aAZtfi)] / 3 (2^)|

(X-p

J

, / (a+0)/2 r^/zA * \2 X Ka-|3 t2Vx 7

>

„ / , (a+£)/2

OJZA'1

*\ 2 X a-pl2l/x)/ 2^ x ^( ^ a +^P V , / / S (^ > f »{ " \ - p 2 ^^ )^}

nbrSs) — ( ^ i . V ^ ^ 1 }

161

CHAPTER IV. GENERAL INTEGRAL CONVOLUTION FOR THE H-TRANSFORMS

As is widely known, famous Laplace, Fourier and Mellin convolutions, which also appeared in the last century, have important applications in various fields of mathematics and physics (for example, E.C.Titchmarch (1937),

I.I.Hirchman

and D.V.Widder (1955),

I.N.Sneddon (1951; 1972).,

N.Wiener (1937), etc.). Except these classical convolutions, there are convolutions

for other

integral

transforms

and

series

in

literature

(N.Ya.Vilenkin (1958), V.A.Kakichev (1967; 1990), M.Flensted-Jensen and T.Koornwinder (1973), I.H.Dimovski (1982), H.-J.Glaeske and A.Hep (1986; 1987; 1988), S.B.Yakubovich (1987), etc.).

Recently S.B.Yakubovich

(1990a) introduced new general convolution

(f*g)(x) which is defined by double Mellin-Barnes contour integrals,who­ se theory being considered in detail in Chapters I and convolution contains a great number of particular cases

II. This general for Mellin type

transforms, the classical Laplace convolution is one of them.

The

basic

factorization

property

for

this

convolution

is

the

following

(K (f*g))(x)=(K f)(x)(K g)(x), 1

where K , K , K

2

3

are some one-dimensional transforms which, in general,

may be different.

In the present chapter we will consider this general convolution in the spaces 3JT (L) and JJ1"1 (L), (introduced in §15) for the case when c, y K , K , K are some H-transforms which are considered in §16. Various examples

of

convolutions

for

the

obtained.

162

classical

integral

transforms

are

§19. Classical Laplace convolution and its new properties

As is known, the Laplace convolution of two functions f and g is defined as follows

(19.1)

For

the

(f*g)(x) =

above

convolution

f(x-u)g(u)du.

there

exists

the

following

factorization

property

(19.2)

(L(f*g))(x) = (Lf)(x)(Lg)(x),

where L denotes

the classical

Laplace

transform

(see V.A.Ditkin and

A.P.Prudnikov (1965))

(Lf)(x)

(19.3)

e

f(u)du.

Now we suppose that f(x), g(x) belong to space !Jft

(L) defined in

c, 7

§15. Then representing f(x) and g(x) by the inverse Mellin transform (15.5) of f# (s) and g* (t), respectively, and after changing of the integration order and using

the beta-integral

(see A.Erdelyi

et al.

(1953) V.l.) we can represent (19.1) in its equivalent form as follows

(19.4)

(f*g)(x) = (2TTI)

2

rd-s)r(l-t) * , *,., -s-t, .. ——r= ^ — f (s)g (t)x dsdt, f(2-s-t) t s

where it is to be remembered that,
t

= 1/2}. For Laplace convolution (19.1) we have the following result. 163

= ^e(t)

L

19.1. The

Theorem exists

for

all

classical

f(x),g(x) € 'JTI

Laplace (L) and

convolution

it

possesses

(f*g)(x) the

(19.1)

factorization

c , "%

property

(19.2). L

— 1/2

In

this

case

-1

(f*g)(x) e !ffi

x

&

if

x~

(f*g)(x)

€ HIT1 ( L ) ,

e JR~ V

y

(L).

Furthermore,

c,min(y+1/2,2^+1/2)

(L) f o r some p a i r i

(c , ^r ) and f o r a i i * *

f(x),g(x)

then jjf 1 c

Proof. From

( D D UJT 1 , 3T

(19.4)

(L).

c,mi n ( y + 1 / 2 , 2 ^ + 1 / 2 )

it

follows

that

the

Laplace

convolution

L

(f*g)(x) exists if and only if

(<*n ^ (19.5)

r(i-s)rd-t) * . * , p ^ —. f (s)g ( t ) € I (.Z-s-t)

Tr X(T ), L((T s t

Using the asymptotic estimation for the gamma-function we obtain (see, also (4.2), (4.3) and (4.10))

(19

'6)

n

r(2-s-t)U "

H

l .. I -1/2 S+ t

>[-5('

exp|-5| |s| + |t|-|s+t,

where E = const, s, t — > oo, (s, t) e {
M O 7^ (19.7)

r(i-s)ru-t) sup (s,t)€((T x cr ) s t

r(2-s-t)

y <

+W

-

On the other hand f(x), g(x) € JJf1 (L) then

(19.8)

expire] |s| + |t| 11 |st|y f*(s)g*(t) e

164

Lir^xcr^

Since 2sgn(c) + sgn(y) ^ 0 (see Definition 15.2),

*

then it follows

*

(s)g (t) (t) e<= L(cr L(cr x x cr cr ), ) , which, which, together together with \ that f f (s)g s

t

(19.7),gives (19.5

L L

It means that the convolution (f*g)(x) exists.

Now by the replacement

x = s+t-1/2

representation

(19.4) can be

written in the next form L

(f*g)(x) (19.9) 2ni

r(3/2-x)

1 ni/2-T+t)ni-t)f (x-t+i/2)g (t)dt, 27ri

dr

where cr = {x € C,Re(x) = 1/2}. According to Definition 15.2 and from L

representation

(19.9)

it

follows

that

x~ 1/2 (f*g) (x) e JJT1 c

(L)

if

and

i ,yi

only if

F(x) r(3/2-x)

(19.10)

e

*VT' *i w , iTl € L ( ( r x } '

where F(x) denotes the inside integral in (19.9)

(19.11)

It

is

F(x) =

evident,

that

27T2

this

r(l/2-x+t)r(l-t)f (x-t+l/2)g (t)dt.

integral

converges for

all

t e cr .

Further,

from (19.10) we have

.

2ni

.

F(x) e r(3/2-x)

_1 271

rn

He T

l

r

, | l x dx

n ^ r M f^>

r(i-s)r(i-t) —^j2 p j —e cr cr t s

71c s + t - 1 / 2

i

y

i . . ._ i i_* . * , , , , ,, s+t-1/2 f (s)g (t)dsdt

165

From (19.6) it follows that the last integral converges if and only if

K -1/2

exp

[«h

>)]

+l/2)|s+t|-l/2|s|-l/2|t||||s+t| *

f*(s)g*(t)

(19.12) <= L(cr x cr ) s

t

Since f ( x ) , g(x) e 311"

(L) then from

(19.8) we have

that

the

relation

c, If

(19.12) holds valid if

{

sup

-i/«i

5

|s+t|

r

|s|_3r|t|"^

*

/

e x p 7r ( C i + l / 2 ) | s + t |

(19.13) -(1/2+c)

s -(1/2+c)

t

I < +00.

By applying Lemma 15.1 to (19.13) we get that the last inequality is equivalent to 2sgn(c -c) + sgn(? -y-1/2 ) * 0, 2sgn(-l-2c) + sgn(-2^) < 0, 2sgn(c -c) + sgn(-l-2c) + 2sgn(* -2^-1/2 ) < 0.

Since

2sgn(c)+

sgn(^)

>

0,

then

the

last

set

of

conditions

can

abbreviated as follows

2sgn(c -c) + sgn(* -y-1/2 ) ^ 0, (19.14) [

It

is

s g n C c ^ c ) + 2sgn(r -2y-l/2 ) < 1/2.

evident

that

conditions

(19.14)

hold

valid

for

c

= c, l

y

= min(y+l/2, 2y+l/2). Consequently,

x" 1/2 (f*g)(x) € 3JT1

#

c,min(y+1/2,2fr+l/2)

166

(L).

be

L

Now let x-1/2(f*g)(x) € ffl"1 (L) for some pair (c ,y ci

), such that

1I

2 s g n ( c ))++ s g n ( * ) > 0. We must show t h a t

m'1

(19.15)

< =11»y » y ii <=

In fact, since defined

by

(L) (L) D m'1

ffl"1

c , m i n (( yy + 1 // 22 , , 223yT++l1//22 ) c,min +1

(L). (L).

C | s l" l 1~ 1_e L(cr )) for for all all ee >>0, 0, then then the the function function h(x), h(x), I e e L(cr

the

inverse

Mellin

transform

from

h*(s) =

x~1/2(h*h)(x) Consequently, we get x"

e-irc|s| ,,-y-i-e» belongs to jjj-i (L) 1 1

(15.5)

c>7

Gm'1

(L). But, as is shown above, this is equivalent to

(19.12),

1 1

* where

* and

* following

definite

form

f (s) =

where f (s) and g (t) have the following |-y-)-a gg.*((tt)) == ee-nc|t| -iic|t|| e-iic|.|| t|-e-i-e -*d.| |s|s-,-)-«a_ -e-i-C> .. ee e |t|

definite

form

f (s) =

[

f (s)

g (t)

have

the

,

71 (Ci+1/2)|s+t|-(c+l/2)|s|-(c+l/2)|t|

€ L(tr L(cr xcr )) s

N-J

|s+t|

7If 1-1/2 -1/2

|st|"^|st|_1" |st|~*|st| ~G

for all c > 0.

t

Since here 2sgn(-l-2c) + sgn(-2*) < 0, then we can apply Lemma 15.3. Hence we obtain inequality (19.13), which is equivalent to (19.14). From the first condition of (19.14) it follows that c

(19.16)

If c

l

ss c. If c

< c, then

2sgn(c-ci) + sgnmin(y+l/2,2y+l/2)-yl ^ 0.

= c, then (19.16) follows immediately also from (19.14). Now from

Theorem

15.1 and

inequality

(19.16) we get

(19.15).

Now we give new

simple proof of the factorization property (19.2). First note that for f(x) € € 5n-1(L) the classical

Laplace

Parseval equality in the next form 167

transform

can be represented

by

(LfMx)

2711

f(s)f (l-s)x ds.

Further, using the representation (19.4) we obtain

- XU .

(L(f*g))(x) =

e

r(i-s)rci-t) r(2-s-t)

1

du

Unir

f

*

. * ., i-s-t,

(s)g

(t)u

,.

dsdt

cr cr t s

r(i-s)rd-t) r(2-s-t)

(2ni)'

f

*

. * , , , ,,

(s)g

(t)dsdt

-xu

e

1-s-t.

u

du

cr cr

t s

r(l-s)r(l-t)f*(s)g*(t)x"2+s+tdsdt Uniy cr cr t

1 2711

s

r(s)f*(l-s)x"sds

-U 27T1

r(t)g d-t)x

dt

= (Lf)(x)(Lg)(x). This completes the proof of the Theorem 19.1.

Now

it

is

to

be

noted

that

representation

(19.4)

is

double

Mellin-Barnes type integral, its integrand involving the beta-function B(l-s,l-t). Consequently, if we replace this beta-function by ratio of two

products

of

the gamma-function,

then we

get

the

new

class

of

convolutions. For example, here we introduce the following definition.

Definition 19.1. The modified Laplace convolution of two functions f(x), g(x) e !JJ1 (L) is called the double Mellin-Barnes type integral

168

(19.17)

(rtg)(x) (f*g)(x) = - i 1 — f [r ) r^f V J J ^^T(s {2ni)2 + t) (2Hi)2

(T
Theorem 19.2. The modifedd all

f(x), g(x) € 3JT1 (L). Then

f'(s)g*(t)x—'dsdt. f*(s)g*(t)x—"dsdt.

j j r(s+t)

Laplace

convolution

x1/2(f*g)(x) € 0JT1

case

there

(19.18)

where A

for (L)

c,min(y+l/2,2^+1/2) c,min(y+1/2,2^+1/2)

c,y c,y

and in in this

(19.17) exisss

is

the following

factorization

property

(A (f»g))(x) (f*g))(x) = (A f)(x)(A g)(x), (A f)M(Ag) lx), +

is a modified

+

Laplace

+

transform

(18.1).

A

This theorem can be proved in the same way as Theorem 19.1. In the following §20 we shall prove more general result. We note that the beta-function B(s,t) = f p ^ /^g^^ ) - is is the the kernel kernel of of the the convolution c (18.1t

(19.17), but the kernel of the modified Laplace transform

is only

onb gamma-functiono

Cohsequenfiy,

here we see ohe

Ao(low1)g correspondence between (19.17) ans (19.18) Ao(low1ng

T(s+t)

<

>

(A+(f*g))

T(s) T(s)

<

>

(A f)

r(t)

<

>

(A + g ).

+

This

simple

consideration

allows

us

to

generalize

convolution

(19.17), where any gamma-function is replaced by the kernels $(x) of H-transform (see (16.8) and (16.9)). This idea initiates new trend in the convolution theory and here we give rather finished results of this class of convolutions. 169

§20.

General integral convolution: definition, existence and factorization property

The main object of the present Chapter is the general convolution which is defined as follows Definition 20.1. Let, as usually, for k=l,2,3 denote

in

ii

w

T-tr(^+b; T]rfr(i-«ik,-!k (20.1)

v

j=l

$ (T)

a.

also ,b.

the be

j=l

^

_

k,

rfr(a; +a-'x)rfr(i-P; -bfx) j=n +i

Let

;

k,

k

complex satisfy

v

numbers the

;

(k)

a.

(k)

, /3

following

J

+iK

j=m

and

the

conditions

real

of

the

positive poles

supposition for the line cr = (1/2-ico, 1/2+ioo) (see Definition

(

Re(/3U))

+

b ( k > 2 > 0,

j

j

(20.2)

k

tfe(a(k))-a(k>2

1 -

> 0,

k = 1,2,3; k = 1,2;

> 0,

j = 1,2,...,n , j = n +1,...,p ,

j

3

b(3)

Re((B\

separation 10.1)

k

> 0,

tfe(a(3)) + a ( 3 ) j

j = 1,2,....n ,

j

Re(oci3))

1 -

1,2,...,m ,

j

numbers

> 0,

*3

j = m 3 + l,,

Then the general convolution of two functions f(x), g(x) e UJT-1 (L) called the next double Mellin-Barnes type integral

(20.3)

(f*g)(x) =

1

r $ (s) $ ( t ) 1

2

$ (s+t)

(ZniV

3

cr cr

t s

170

f (s)g (t)x" s

dsdt.

is

First we establish the conditions for the existence of convolution (20.3). According to (16.11) and (16.12) denote

m (20.4)

K

~

k

n

k

u(k) +

) b

2

j=i

Pk

(

(20.5)

„

/u = Re

„

k

; a

p (k)

„

->

j=i

a

(k)

, (k)

b

q

)

j=i

k

Y

j=m +i k

p k ^

„ ^(k)

„

-

j=n +i k

J=i

qk (k)

q

k

*k

- v b

j

j=i

p - Mq

k „ , (k)

(k)

a

k

j=i J

k = 1,2,3.

Then from the asymptotic estimation for the gamma-function (4.2) we obtain

$ (s) $ (t) 1

2

exp ir k

| s+t | -K |S|-K lt|

$ (s+t) 3

(20.6)

-v

M

Is+t | 3 | s | where E = const, s,t

a

-y |t|

2

,

> oo, (s,t) e (cr x(r ) and s

(

(20.7)

P

t

q

3

~

(3)

j=l

3

„ , (3) j=l

Note here that (20.6) can be followed immediately from the asymptotic estimation (11.3).

Theorem f(x) e m'1

20.1.

The

general

(L), g(x) € m'1 1 1

convolution

(L) if 2

2

171

(20.3)

exists

for

(

2 s g n ( c +K - K ) + sgn(3r + ji -JJL ) ^ 0,

1

k

k

3

k

k

k = 1,2;

3

2 s g n ( c +c +K +/c ) + sgnt^r +^r +JLI +JI ) ^ 0,

(20.8)

s g n ( c +K - K ) + s g n ( c +/c - K ) + s g n ( c +c +K +K ] &

1

1

&

3

2

+ 2sgn(yi+r2+fii+^2--fi3)

1/2

In this case defined

as

1

2

1

2

—1

the

pair

(c ,^r ) is 3

3

follows

'

1

1

3

1

1

),

if

3

1 if

1 *1 * 3

2 ^2

I VVWV''

2

2

if

^3

C +K 1 1

C +K . 2 2

T h i s theorem can be p r o v e d by a n a l o g y w i t h Theorem 1 9 . 1 f o r

Laplace convolution. convolution

1

C +/C < C +K ; 2 2 1 1

(c +K - K , min{^ +U. -u , ? +u -u 1 1 3

Proof.

c +K < c +*c :

3

(C +K ~K , 3T +/J. -Lt ) 2 2 3 °2 * 2 ^ 3

(c , y ) 3

(20.10)

&

3

5: 0.

(f*g)(x) 6 DTI (L), where Vr3

x

( (c +/c -K. , 3r +U. -u

(20.9)

2

In f a c t ,

(f*g)(x) e x i s t s

|s+t|

3

Mt1

|s|

2

from

(20.3)

and

(20.6)

we h a v e

that

the

if

exp^/c3|s+t|-Ki|s|-K2|t|jlf*(s)g*(t)

€ L(cr xcr ) . s

t

1

S i n c e f ( x ) € DJf ( L ) , g ( x ) e JJJf1 (L), c ,7 c ,? 1 1

(20.11)

f (s) e

2

Isl

ii

i.e.

2

€ L(crs ), °g (t) e

*

Itl

ii

2

€ L(crt ),

therefore relation (20.10) holds valid if

sup

{|s+t|R3

1 "l

2 ^2

(s,t)G (
s

t -(C +K ) t 2 2 ' '

< +00.

172

'['h

exp|7T|K I S + t I -(C +K )|s|

But this is equivalent to conditions (20.8) by using Lemma 11.4.

Let x1/2(f*g)(x) 6 JJf1 (L) for some pair (c,y). Then from (20.3) we c,7

1/2

represent x

(20.13)

(f*g)(x) as follows

x"TF(x)

x1/2(f*g)(x) = -2niU to

$

dT

(T+1/2) 3

where

(20.14)

F(T) =

$ (x-t + 1/2)* (t)f (x-t + l/2)g (t)dt for T 6 (r .

2ni

1

2

T

1/2

Hence, according to Definition 15.2 x

-1

(f*g)(x) € !JJ1

(L) if and only

if (20.15)

g H/2-HT)

e7rC

'X||Tr

G

L(l/2-ioo,l/2+ico).

3

Using the equivalent symbol = of simultaneous convergence or divergence of two integrals from (20.13) and (20.14) we have

F(T) (1/2+T)

$

7TC|T| I | y ,

e

x

dx

3

$ (s)$ (t) 1

2?r

2

71c | s+t-l/2|

$ (s+t)

s+t-1/2

f*(s)g*(t)dsdt

3 t s

| S+ t |

|s|

It I

^ e x p 71 ( c + K 3 ) | s + t | - K j s | - K

( T CT

t

s

x |f (s)g ( t ) d s d t |

173

2

|t

)]

Since

(L), g(x) e Ml"1

f(x) e Ml '

(L), then

the

last

integral

converges if

(

sup

(s,t)€(CT xtr ) 1v s

s+tl

-V M 2

-M

Is

111

r r exp 71 (C+K ) |s+t I

t

(20.16)

' ( C l + I C l ) l S l " ( C 2 + l C 2 ) l t l ) ] } < + £ °Again using Lemma 11.4 we get that (20.16) holds valid if

f

2sgn(c+K - c - K ) + sgn(3r+/i -y

1

3

k

k

3

- J I )< 0, k

k = 1,2;

k

2sgn(-c i -C 2 -K i -K 2 ) + sgn(-'yi-2r2-fii-M2) * 0, (20.17) sgn(c+K -c -K ) + sgn(c+K -c -K ) +sgn(-c -c -K -K ) & 3 1 1 3 2 2 ° 1 2 1 2 + 2sgn(^+M3~^1-?'2-fi1-^2) s 0.

Now it is easy to verify that the last set of conditions (20.17) holds valid for the pair (c.^r) = (c ,v ), which is defined by (20.9). It means 1/2

that x

-1

(f*g)(x) e UJ1

(L) (here we also use (20.8) as the existence

conditions for the convolution (20.3)). Theorem 20.1 is thus completely proved.

■

In §16

we defined the H-transform of function f(x), belonging to

!JJ1 (L). Here we give definition of this transform of function f(x), such that x A f(x) € 3n_1(L)f A 6 (R.

Definition 20.2. Let A be a real constant and a function x f(x) € JH (L),i.e. it is represented in the next form

(20.18)

x A f(x)

1 2ni

f*(x+A) x"TdT,

where f (T+A) e L(cr ). Then the H-transform with the kernel $ ( T ) of the function f(x) is interpreted as follows 174

$(x+A)f*(x+A)x~Tdx = - J^ U U 2ni

(Hf)(x) = -%-A ^j 2ni

(20.19)


$(x)f*(x)x~Tdx.

Re(T)=

— +X 2 2

We n o t e t h a t i f $ ( x ) f **x) i s an a n a l y t i c f u n c t i o n i n some s t r i p

8,

invoWeino t h tthat v e r tif i c a$(x)f l l i n e(T) s fti(xa 1/2 and function J?e(x) = 1/2 + A, strip moreover We note is ann analytic in some 8, thvo vingtion vs i n t e g r lines l b l e on any n v1/2 e r t i c aand n J?e(x) l e n e o=f 1/2 th++ A, s t mmoreover i p 8 and invoWeino tht vertical fti(xa t(x)ff(xc x € 8e on |l/n(x)| thvo vingtionn ivs for integrlble any vertican t(x)ff(xc $he $he the the

ni

lene of

th+

stmip 6 and

for T € 9e |l/n(x)|

^ oo then from hauchy ipeorem ^ oo then from hauchv ipeorem f-transform (20.19) ,oincides wita ,ht H-transform defined by f-transform (20.19) ,oincides wita ,ht H-transform defined by 8) whose integral concour id the ithe
(

According According result. result.

to Definitions to Definitions

Theorem 20.2. Let kerness

2

(20.20) (20.20)

20.2 we 20.2 we

on

the

k

1

g ( x ) € JJf V

k

k

convolution

(L) if

y

following following

(H2g)(x) with UK'1

spaces

c ,y l1 i1

the

(L) and

and only

k = 1,2.

*k

( 2 0 . 3 ) exists

for

all

f(x)

€ JJJf1

(L),

if

2 f

2sgn(c +K -K ) + sgn(y + n - / )

\

sgn(c +K -K ) + sgn(c +K -K ) + sgn(c +c +K +K )+

&

[[

k

1

I

k

1

3

k

3

2

*k

2

^3

£ 0, k = 1,2; 2: 1

3

andt and,

2

1

2

+ 22sgn(r -^) + s g n ( r i +1y+r ++fi M 2 -2M 3 ) ** o. 0. 2 + (2x^i 1

(L), where In this case x1/2(f*g)(x) € JJf JJT11 (L), Vy3 (20.9),

the the

iie.

2 s &&g n ( c +/c +*c ) + s g n ( y +n ) 2: 2: 0, 0,

Then the general

obtain obtain

(H^Mx) and (H^)(x)

exitt

$ (x), $ (x) (20.1) 1

and and

H-transforms

m'1 (L),respeceively, V°2

(20.21)

20.1 20.1

moreover,

and g ( x ) e€ Jm" and JJf11 (L), V*2

c2,r2

if then

(c ,y ) is defined 3 3

x ) e JJf 3JT11 (L) for x 1 / 2 ( f «* g )) ((x) f o r aalli i f ( x ) c.r UJf1 (L) D JJi JJf1* JJf c'* V

cr

y

(L). 3

c 3 ,r 3

175

by

JJT1 (L) e€ 9H" ci>yi

Finally, H -transform 3

there

is

(H f)(x) with

the

following

the kernel

factorization

$ (T) for

3

property

the convolution

3

(20.3) by the product

°

of two H-transforms

(Hf)(x) and (Hg)(x): 1

(20.22)

of

(f*g)(xj

2

(H (f*g))(x) = (H f)(x)(H g)(x) 3

where the left

Proof.

part

of

1

2

(20.22) is understood

by Definition

20.2.

From (20.20) it is evident that 2sgn(c +c + K + K ) + sgn(y +^r +u +u ) £ 0. &

1 2

&

1 2

1

2 'l * 2

Hence according to the set of conditions (20.8) in Theorem 20.1 we see that inequalities (20.21) together with (20.20) provide the existence of 1/2

the convolution

(20.3),

and we have x

—1

(f*g)(x) G 3J1

(L) in this 3

3

case , where (c ,y ) is defined by (20.9). 1/2

We shall show that 5JT1 (L) D JJT1

(L). Indeed, we set

-7Tc Is | -y -l-e # f (s) = e |s| ,

Then according to definition of !JJ1

all e > 0.

Consequently,

(20.15) holds for the function F ( T ) ,

176

It | 2

-r -i-e Itl 2 ,

e > 0.

(L), we have that the corresponding

from

form

-TIC

0

g (t) = e

functions f(x) and g(x) belong to 3JT1 for

-1

(f*g)(x) e 5J1 (L).

Now let there be some pair (c,y) such that x

1

(L) and HIT2

(20.13)-(20.15)

2 (L)

respectively

it follows

that

which has the following definite

1

F ( T ) = ^ij exp|-7r|c |x-t + l/2|+c |t| 11 |x-t + l/2| (20.23) x

dt.

t

In this case using the replacement x-t+1/2 = s and the equivalent symbol = of convergence or divergence of two integrals from (20.23) we obtain

F(x) $

^irclTL.y

dx

(1/2+T) 3

|s+t|

|s|

|t|

exp|7i|(c+K )|s+t|

[■('

(T (T t s -1 -£ "(C+K )ISI~(C +K ) t l l 1 1 2 2 '

IstI

Idsdtl

Now using conditions (20.20) and applying Lemma 15.3 we get that the last integral converges if and only if inequality (20.16) holds valid, i.e. we again obtain conditions (20.17). Here conditions (20.20) allow us to abbreviate (20.17) to the following conditions

2 s g n ( c - ( c +K - K )) + s g n ( ^ - ( ^ + [i -[i )} (20.24)

s g n ( c - ( c +K - K ] &

0, k = 1,2;

+ s g n ( c - ( c +K - K )) - s g n ( c +c +/c +/c )

1 1 3

2

2

3

1

2

1

2

+ 2sgn(2r-(*1+*2+fi1+^2-M3)) < 0.

Now for any pair (c,^), satisfying (20.24) we must show that JTJ

(L) D

c , 'y -1 (L), where c > „ c3 ,2"3

JT1

(c ,y ) is defined by 3

(20.9).

3

first inequalities of (20.24) it follows that 177

Indeed, from

the two

c ^ min{

But

from

(20.9) we have K3>c2+*2-K3}} 'VV+VS*}

min{c + Kmin{c + Km X1 (L) z> m'11 c , y (L) z> m'c 5JT c y

Now Now

C + K - K , 1 1 3

inequalities

3

i i

= min{c + K -K , c + K -K }} .} Hence 1 1 3

follows follows

2

from from

2 K

ii c ^

3

properties properties

((15.12 ((15.12

that that

two two

ff ii rr ss tt

(L) for all y, y e R. (L) for all y, y3 e R.

y

c3>9r3

ll ee tt

c

C + K - K } . 2 2 3

cc = = cc

3

= +/c -K - K }>.. = min{c min{c +K +K -K - K ,, cc +/c

3 3

11 11 33

22

22

Then Then

3 3

m'1 ( 2 0 . 2 4 ) we have IJJf 3JI *1 (L) D :> JJf1

from from

_

* ((L), * L ) , k = 1,2. 1,2.

+ K K. , y +LlLl LlLI cc k +Kk K3 , y k + * k 3

cc ,,yy

1 1 Consequently, according to (20.9) we must prove t h a t m lUT (L) z> UTf1

c ,y 3 3

c,r c,y

in the case c +/c = c + K , i.e. , when in the case c +/c = c + K , i.e. , when 1 1

1 1

2 2

2 2

(L)

c = c = c + K -K = c + K -K . In c = c = c + K -K = c + K -K . In 3 3

1 1

this this case case from from the the last last inequality inequality of of (20.24) (20.24) we we get get

1 1

3 3

2 2

2 2

3 3

2 s &g n ( y - ( y +y +fi +JLI - / ) ) < tos g n ( c +c +K +K ) < 1, &

11

to

22 **1 1 **2 2 *^3 3

1 1 2 2 11

y y +y +LI +M +LI +jn -LI\ -jn*. Hence H e n c e Mf UJT11 (L) (L) 3 3 3 IJJf1 y ± ^ y 11 *

i . ee..

C

'*

°3

2

(L) for f o r any a n y pair p a i r (c,y), (c,y), (L) 3

satisfying (20.24). Further, as it is shown above, we have x

(f*g)(x) <= JT Jl c

Using

representations

(20.13)-(20.14) (20.13)-(20.14J

for

the convolution

(L). 3>3r3

(f*g3(x),

Usingding ro Definition (202 we obtain (H3(f*g))Hx) =

tio/2

r

^nT

U ( x + 1 / 2 ) $ (TIi/2)x"Tdx

J

cr

-

T

= ^ _JL _ _2 f\ $

(2*i) J '' (2ni)2JJcr J cr

T T1/2 -t+1/2)* (t)f*(x-t+l/2)g*(t)x__1/2 dtdT (T-t+1/2)* (t)f*(x-t+l/2)g*(t)xdtdT

cr cr Tx tt

=

^ _ [ $ (s)f'(s)$ (t)g*(t)x-s-tdsdt (27ii) 2 J J 1 2 cr cr s s t t

= (H f)(x)(H g)(x). 1 1

2 2

178

Here the existence of the H-transforms (Hf)(x) and (Hg)(x) is provided by inequalities (20.20). Theorem 20.2 is completely proved.

We

defined

the

general

convolution

as

double

■

contour

integral,

however, under some additional conditions it can be represented by a real form. The following theorem is true.

Theorem 20.3. Let

the

parameters

, (3

a j

conditions

(20.2). Let

also

the

following

, a

j

satisfy

, b j

the

j

inequalities

hold

2sgn(fc - K ) + sgn(/i -JLI - 1 ) > 0, k = 1,2; k

3

*k

*3

2sgn(K + K ) + sgn(u +u -1) > 0, 1 2

(20.25)

1 2

2min-jsgn(K - K ) + sgn(/c -/c ) , sgn(/c -/c ) + sgn(/c +K ) , 3

2 3

sgn(K -K ) + sgn(

Then

the

general

convolution

1

1 3 2

/

can

(20.3)

1 2

+ sgn(jj. +M ~M ~2) > 0.

be

represented

2

integral

on R

as

follows

H[£ .5]f( U )g(v: dudv uv

(f*g)(x) =

(20.26)

o o where

H [x,y]

H [x,y]

is

H-function

of

two variables

$ (s) $

1

1

$

(2711 ]

2

(t)

-s -t ,

x

(s+t)

of

y

type

(10.1!

,.

dsdt

3 cr (T t s

p -n (20.27)

*3

, q -m ; m , n ; m , n 3

q ^3

3

»

3

p *3

1

1

2

; p >q ; p >q M

^1

^ 2

179

n

(/3(3),b(3))

2 x,y

2

,

(a

m +1 , q 3 3

(3),

(3)

,a

n +1, p 3 3

by

double

ro(3)

(/3 r

(3)

(a

,

, (3K

,b

(3)

,a

)

1 ,m 3

^

(1)

: (a

rod)

)

(/3

(1).

,a

1 , p 1

In fact, from

(2).

; (a

;fo(2)(/3

,(1K ,b )

1 , n : 3

Proof.

, (2)

)

,a

, (2K

,b

11.1

it

]

1 >P 2

)

1 ,q 1

Theorem

x

)

1 ,q 2

follows

that

the

double

integral in (20.27) converges if and only if conditions (20.25) hold. In this case it is evident, that

$ (s) $ (t) SUp (s,t)€((T X(T ) s t

1

2 (s + t )

$ 3

Hence convolution (20.3) exists for all f(x),g(x) e 9JI (L).

Further, conditions H-function

(20.27).

convolution

can be

(20.2) are required for the existence of the

Finally

representation

transformed

from

(20.26)

(20.3) by

of

the

the using

general

the double

Mellin transform of H-function (20.27)

(20.28)

m 1 H [x,y];s,t}- = 1

'

$ (s) $ (t) * ?— , (s,t) € (
(S+t)

s

t

(see also §7, Theorem 7.3). Theorem 20.3 is thus proved.

We conclude by remarking that if $ (s+t) = 1, then the convolution (20.3) is the product of H-transforms (Hf)(x) and (Hg)(x), i.e. in the present case we have

(f*g)(x) = (H f M x M H g)(x),

180

if

* (s+t) = 1.

§21.

According

Typical examples of the general convolutions

to Definition

evident that convolution

20.1

of

the general

convolution,

it

is

(20.3), generally speaking, does not possess

the commutation property, i.e. (f*g)(x) is not always equal to (g*f)(x). It is not difficult to note that (f*g)(x) = (g*f)(x) if $ (T) s $ ( T ).

In this section we consider only symmetrical cases of the general convolution, when $ (x) = $ (x) and both functions f(x) and g(x) belong to JJ]

(L). Some typical examples are also obtained. For convenience we

c, y denote $ (x) = $ (x) = <Mx) and 1 2

Theorem 21.1. The

(21.1)

(K , u ) = (K ,LI ) = 1 1 2 2

(K,U).

convolution

$(s)$(t) *

(f*g)(x)

$

2

(27ri)

(s+ t

)f

. * ., -s-t, ,.

(s)g (t)x

dsdt

3

exists

for

all

fix),

(L) if

g(x) e 3T1

and only

if

c,7

2sgn(c+K-K ) + sgn(^r+fi-|Li ) ^ 0, 2sgn(c+K) + sgn(3r+/ii) ^ 0,

(21.2)

2sgn(c+K-K ) + sgn(2y+2^-/ui ) ^ 0.

In

this

case

(f*g)(x) = (g*f)(x)

and x 1 / 2 ( f * g ) ( x )

e JJl * ( L ) , c , If 3

= C+K-K , K = mint^+^i-ji ,2y+2n-iA For convolution

(21.3)

Moreover,

(21.1) the

); K , [i factorization

are

defined property

(H3(f*g))(x) = (Hf)(x)(Hg)(x)

if 181

where

c 3

3

by (20.4),(20.7). holds

2sgn(K-K ) + sgn(/n-/n -1) > 0, 2sgn(K) + sgn(2fi-l) > 0,

(21.4)

2sgn(/c-/c ) + sgn(2/i-jLi -2) > 0,

then

the convolution

(21.1) exists

can be represented

as

(21.5)

(f*g)(x) =

for

all

f(x),

(L) and

g(x) € JH

it

follows

dudv H [[u- ,-lf(u)g(v: & vj uv

where

H [x,y]

<S>(s)$(t) -s -t , .. —=-—-, r-T- x y dsdt y $ (s+t)

1 {2niV

cr cr t s

(/3(3),b(3))

m +1 ,Hq 3 3

p -n ,q -m ; m, n; m, n 3 3 ^3 3 x,y q 3 , P 3 ; p>q;p,q

(21.6)

f

(3) ( 3 ) .

(a

,a

)n +1, p 3

(/3(3),b(3))

:(cc,a)

;(a,a)

1,m '3

(a(3),a(3))

Proof. ig taking

1,p

(|3,b)

1, n : '3

;(/3,b)

1,q

3

1 1 ,p

1,q

y

According to Theorem 20.1 of the convolution existence,

( y )) = ( y )) = (c,y) (cc ,,y (cc ,,y (c,^)

and and

(K ,JH ) = (K ,/I ) = (K,JLI)

(20.8), we get the following conditions

2sgn(c+jc-K (21.7)

) + sgn(?+/a-/j. ) ^ 0,

2sgn(c+K) + sgn(2r+/i) > 0, 2sgn(c+K-K ) + sgn(c+/c) + 2sgn(23r+2fi-fx ) > 0 .

182

from

From the two first inequalities of (21.7) it follows that C+K-K

^ 0,

C+K ^ 0. Hence in this case it is evident that the last inequality out of (21.7) is equivalent to the last one out of (21.2).

Consequently,

(21.7) can be written in simpler form than (21.2).

Now according to (20.20), the second

inequality out of

(20.21), and applying Theorem 20.2 from (21.2) it follows that the convolution

(21.1) exists for all f(x), g(x) e 3JT1 (L) if and only if the set of conditions (21.4) holds valid. Further, in accordance with (20.25) from Theorem 20.3, the double integral in (21.6) converges if and only if

2sgn(K-K ) + sgn(n-^. (21.8)

-1) > 0,

2sgn(/c) + sgn(2/n-l) > 0, 2min-{2sgn(K-K ), sgn(K-*c ) + sgn(

"}

+ sgn(2M~fi -2) > 0.

Here two first inequalities of (21.8) show, that K ^ 0, K-K £ 0. Then

*

the last inequality from (21.8) means that the condition 2fi-/Lt -2 > 0 is required if and only if K-K

Hence it is not difficult to obtain

that (21.8) is equivalent to (21.4). Thus Theorem 21.1 is proved.

Now we consider some original examples of convolution (21.1).

Example 21.1. Let <Mx) = T(x) and $ (x) = T(x)/k (x), where k (x) is Mellin transform (15.3) of some function k(x). Then the corresponding convolution can be written in the form

(21.9)

r(s)ru) * _,_.,*, x *,., -s-t, ,.

(f*g) (x) ( 27T 2* ]

r(

+t)

k (s+t)f (s)g (t)x

dsdt.

At first we propose that the function k(x) is Fox's H-function (1.5) and * k (x) is defined by formula (16.9) 183

m n "FT T O +b x) F T r(l-a -a x) j

j

p FT

j

i-l_

q r(a +a x) F T r(l-j3 -b T ) J

j

j=n+i

j

j

j=m+i

According to formulas

(20.4),

ji = 0 and

21.1

by Theorem

j

J=1

k*(T) = -^

existence of the convolution

(20.7) in our case K = 1/2,

(20.5) and

it

is not

difficult

to obtain

that

(21.9) for all f(x), g(x) e 3JT

the

(L) is

* * provided by the next conditions (here K ,a for $ (x)=r(x)/k (x)) 3

3

3

* 2sgn(c+l/2-K ) + sgn(y-ju ) £ 0,

(2i.io)

3

3

;

^

2sgn(c+l/2-K ) + sgn(2?-^F) * 0.

Further the factorization property (21.3) in this case can be obtained by the aid of composition structure of the corresponding H-transforms. According

to

formula

(18.1)

(Hf)(x) = (Af)(x)

and

since

$ (x) = 1

T(x)/k*(x) then by the Theorems 16.4 and 16.5 (H f)(x) = (A oK"" ) (f (x)) , where the H-transform (K f)(x) is the inverse transform for (Kf)(x) * with kernel k (x). Hence the corresponding formula (21.3) takes the next form

(H (f*g) )(x) = ((A o K 1 )(f*g) )(x) = (A f)(x)(A g)(x).

(21.11)

3

From conditions existence

of

real

+

+

+

+

(21.4) it is not difficult representation

(21.5)

it

is

+

to see that for sufficient

that

the the

following inequality be valid

(21.12)

2sgn(l/2-K ) - sgn(fx +2) > 0.

However, in this case the H-function of two variables

(21.6) can be

evaluated by the reduction property (12.3), formula (20.28) and the next identity (see

Nguyen Thanh Hai and S.B.Yakubovich (1990)) 184

r(s)nt) * r(s+t) l

(21.13)

, , J

^ s-l

k(x+y)x

t-1 , .

y

dxdy.

o o

«N -

Hence H|^,^| in (21.5) is equal to k

and we obtain the so-

called sum-convolution

(21.14)

The

k X

[ ^]f(u)g(V

(f*g) (x) =

identity

(21.13)

is

true

dudv uv

if k(x) € L(0,oo),

therefore, we

can

consider the convolution (21.9) in form (21.5) for arbitrary kernel k(x) € L(0,oo). For this convolution it not difficult -1/2 to get the norms estimations in L-spaces. Thus, if f(x), g(x) € L(x

;0,oo) from the following chain

of inequalities 00

03 00

dx

*H?] f(u:

dudv g(v)

uv ,_ . , .,dudv — — f ( u t) g ( v )

|k(x)|dx

i4-\/

I

°

rig(v)

rlf(u) |k(x)|dx

I in;

dv,

-du Vu

Vv

we have

(21.15)

|(f*g)J

^ M|fI L(0,oo)

where M

_x L(x

;0, oo)

m L(x

;0, oo) '

|k(x)Idx.

Many examples of the sum-convolutions can be obtained by the Table 13.1. Let k(x) = 2K (ZVx~) o

is Macdonald function 185

(see line 47 in the

Table 13.1). Then the corresponding sum-convolution (21.14) and factori­ zation property (21.11) take a form

(f*g)+(x) = 2 (21.16)

K

of*■ 2 /^vl' f ( u ) g ( v )uv ^X •

(A^tfsg) )(x) = (A f)(x)(A g)(x).

1 Example 21.2. If $(x) = 1/T, $ (x) = — , then similarly from 3 xk (x) the representation (21.9) we obtain the so-called max-convolution

(21.17)

(f*g)

k x max{u" ,v~ > f(u)g(v vdudv

(x)

It is not difficult to see the corresponding conditions of existence from (21.2), (21.4) by K = 0, JI = 1. The H-function (21.6) in this case can be evaluated by the maximum reduction property (12.4) and it is equal to k x max{u ,v > . Further, the factorization property (21.3) takes the next form (H (f*g)

)(x)

3

(21.18)

= ((I x_1)o K _1 )(f*g)

)(x) = I1(x"1f(x))I1(x"1g(x)),

max

-

-

where the simplest operator I (x"f(x)) is defined by Table 17.1 (line 7).

As — 1/2

L(x

in case of sum-convolution, if k(x) € L(0,oo), f(x), g(x) € ;0,oo) then for max-convolution (21.17) the norms estimation is

also true

(21.19)

||(f*g)oaJ

*2M|f| L(0,oo)

L(x

where constant M is defined by (21.15). 186

|g| ;0,oo)

L(x

;0,oo)'

Actually, 00 00

CO CO CO CO

kfx max{u" 1 ,v" 1 }]f(u)g(v)^X dx II JJ uv 00 00 0

oo oo

**

oo oo oo oo

|k(x)|dx

min{u,v>|f(u)g(v)|^X min{u,v>|f(u)g(v)|^^

00

00 0

oo oo

oo oo oo oo

==

|k(x)|dx |k(x)|dx ' '

1

o

<<

rif JJ

oo o

|k(x)|dx |k(x)|dx

00

— — ^^ -|f(u)g(v)|^X && max{u,v>' '' uv uv max{u,v>'

nrif(u)i f(u)i

rig(v)i 7ig(v)i

du du

dv. dv.

J J vv ^^ JJ v'v" v'v" 00

0 0

According to Table 13.1 (line 48) the corresponding max-convolution (21.17) with Macdonald function K (2Vx~) (2Vx"~) is the following o oo 00

(21.20)

(x) (( ff *«gg))r M o aX x(x>

= 4z\ =

oo 00

1

|[ K K Oo(( 22 // xX mma ax x< {u u- - ,1v, -v1- > 1 )>f)(fu()ug)(gv()v^)X^ ..

00 0

The basic factorization property (21.18) for this convolution takes the the form

(21.21)

(2Vx)\ 1o[(f*g) °|"(f*g) (u)l J2x1/2K (2Vx)\" (u)l I III max JJ 1 I max

=

( x - 1 f(x)) ( x ) ) II1^ (x^gtx)), (x-1g(x)) II1 ^(x'V -

1/2 where operator of the inverse Meijer transform JJ2x KK i(2i/x)l 2 x1/2 ( 2 v ^ ) r 1 o°|f(u)| rf(u)l

is defined by Table 17.1 (line 25). 187

Now we give the table of the kernels for sum- and max-convolutions (21.14), (21.17) and the corresponding H-transforms in the left parts of the factorization equalities (21.11), (21.18). This table is based on the Table 13.1.

TABLE 21.1

X+y kf(max(x,y)J ]

1

2

(H3f)(x)

e-x-y

E (x A ' V 1 )

exp - m a x ( x , y )

+

3

r(a)(l+x+y)"a

4

T(a)|l+max(x,y

5

nsy

(1 x

- -y } +

6

f

7

Eij-x-yj

8

9

^y(l-max(x,y

(

(x

>r

|r(l+a)x(l+x)"1"a|

l

(x A x

)

+ , a i - a -lx

(x TI_

•):'

x

)

[■XT' [IV^IA;1)

Ei - m a x ( x , y )

Vn

- a A - i a. A x )

l1/2x-l/z)

(x

erfc{Vx+y)

t 1/2.-1 -1/2N

10

Vn e r f c ( ^ m a x ( xTyT)

(x

11

T(v,x+y)

(x I

12

f[v,max(x,y)]

(x A x

A x

X

+

188

)

+

) )

13

f-'-^r2]/,

J42V^)

x (l+v/2;l+v;~)|

14

( ,

^

{x—r[--^-] iF2 ( 1+ , /2;

Jv^max(xfy)J

2+v/2,l+v;~)|

|x- 1/2 e- x/2 W

15

16

^

i

■■[:]/,[: H

18

■"['] ,Ft[i I""""* "]

20

r[V] /,[•;'

J

(iV 1 )* J2K^(2^)|

2 K^ 2^max(x,y)

17

19

(x)V 1 1/2,IV2

(x

1—a T a-b b - 1 x I X ) +

H^V 1 ^ 1 * 13 ^} 1 f e -l/(2.) (l)!"1 | 1-c,(a-b)/2 X J

-x-y

|xd + x) c - a - b - 1 r[ 1 ^; 1 + b ]

r[V]/JV | -max(x »]

x F (c-a,c-b;l+c;-x)\ 2 1 J -l

21

r(a)T(a-b+l)

22

r(a)r(a-b+l)

1

tf|a,b;x+yl

b

b

a

1

|r(l+a-b)x " (l+x) ~ " j

tf|a,b;max(x,y)l

189

|xr(l+a)r(a-b+l)*(l+a,l+b;x)i

§22. Case of the same kernels: the general Laplace convolution

In this section putting $ (x) = $ (T) = $ (x) = <Mx) in Definition 20.1

we obtain

corresponding convolution

the so-called

algebraic

classical

properties.

is the modified

case

of convolution

The simplest

Laplace

convolution

example

(19.17).

difficult to show from the Theorem 20.3 that the integral convolution

(19.17) diverges. But we can bring

with

of such It is

not

(20.27) for

the modified

Laplace

convolution to the similar form (19.1). Our new approach to convolution constructions naturally allows to generalize convolution

and obtain

I.H.Dimovski

the known examples

and V.S.Kiryakova

the classical notion of (see I.H.Dimovski

(1984,1985),

V.S.Kiryakova

(1974),

(1989)) as

well as the new ones with the corresponding factorization properties. Earlier these constructions were used in operational calculus. The known ways of constructing

the operational

theory are also applied

to

the

objects considered. Now we establish the main theorem for convolution (20.3) in the case of

the same

simplified.

We

kernels.

The corresponding

denote

(K ,/LX ) = (K^,/^J U 2'fl2)

conditions

accordance with (20.4), (20.5) m [j=i

n

p

q

j=i

j=n+i

j=m+i

P

q

j=i

j=i

p

Theorem 22.1. The

(22.1)

(f*g)(x) =

J

q

p - q

Ea - Eb

fi = Re

convolution

$(s)$(t) »(s+t)

1

(2rrir (T (T t s

where

190

* f

are greatly

= (K^,/I„) = (K,/I), = U 3'^3) = ( K ^ ] '

. *

, - s - t , ,. d s d t '

( s ) g U ) X

where

in

f r r ( V V ) rTr(i-aj-ajT) (22.2)

<Mx) = 7 7 rfa.+a.x] 7 7 rfl-p.-b.r)

and the parameters

satisfying

,

the following

Re{(B ) + b / 2 > 0, J J

j = 1 , 2 , . . . ,m

l-/te(a

j = 1,2,. . . , n

) - a J

(22.3) j

j

■ l-/?e(3 ) - b

for

all

fix),

g ( x ) e 5TI

> 0, j

Ke(a ) + a

exists

conditions

> 0,

j = n + 1 , .. . , p

> 0,

(L) if

j = m + 1 , .• -q

and only

[ 2sgn(c) + sgn(y-

P q Ea - £ b

-

j=i

P

q

J=l

J=l

E a - Ib

2 s g n ( c ) + sgn(2r+M~2

1/2

x

(22.5)

For convolution (22.6)

o,

j=i

2sgn(c+K) + s g n ( r + / - t ) - 0 ,

(22.4)

i n t h i s case

if

-1

*

( f * g ) ( x ) € !JJ1

".

) * 0.

, (D> where y

=

2

= min(2r-jm#,2y+jLi-jLi#) and

p q [ a - [ b

(22.1) the factorization

property

(H(f*g))(x) = (Hf)(x)(Hg)(x).

Moreover, if 191

holds

2sgn(K) + sgn(2fi-l) > 0, (22.7) fi# + 1 < min(0,Li-l),

then

the

convolution

can be represented

(22.1) exists as

all

(L) and

f(x), g(x) e 5J1

H[n >^u)g(v

(f*g)(x) =

(22.8)

for

it

follows

dudv uv

where

*(s)*(t) -s -t , ,, —^ ^j— x y dsdt y $(s+t)

1

H[x,y] =

Uni)'

cr cr t s

(22.9)

O.b) p-n,q-m;m,n;m,n

=H

q ,P

,(/3,b) m+l,q

: (a,a)

1,m

,(a,a)

n+l , p

: O.b)

;0,b)

1, n

1 ,q

1

Proof. According to * (K , a ) = ( K , U ) , LI = Lt+Lt , 3

1, P

x,y

;p,q;p,q

(a,a)

3 3

)

;(a,a)

1,p

Theorem where ix

1

1, q

2

2

21.1 taking (K ,U ) = (K ,U ) = i s d e f i n e d by ( 2 2 . 5 ) i t i s n o t

*

*

difficult to obtain from conditions (21.2) the equivalent inequalities (22.4). Further, since K in (21.4) hold

= K, then the first and the last inequalities

if fi#+l < 0 and LI-LI#-2 > 0 and we get

(22.7). Thus Theorem 22.1 is established.

Remark

22.1. For

modified

Laplace

LI = 0, LI = -1/2 and we obtain only

the conditions

■

convolution

the conditions

(19.17) K = 1/2, (22.4) for

this

convolution (the conditions (22.7) do not hold and the integral (22.9) diverges). Now we consider for convolution (22.1) some approach of constructing the integral representation as a generalization of the classical Laplace integral

(19.1) which

is based

on

192

the composition

structure

of

the

H-transform r

,

. s

(see §16). As we noted above, if we set <Ms) = r(s), then

= B(s,t) is the classical beta-function and the corresponding

convolution (22.1) is the modified Laplace convolution.

s Definition 22.1. We shall call the kernel *[$(s+t) f * * f ^ in (22.1) as the

general beta-function and denote it by B(s,t). The convolution (22.1) is naturally called the general Laplace convolution.

Next, we use the value of the beta-integral

(22.10)

B(s,t) =

u

(1-u) du

and we represent the general beta-function B(s,t) as follows

\~\

r|2|5.+b.(s+t)|

7^"r|2-2a.-a.(s+t)] 1 *(s+t)

8(s,t)= "TTrfa +a s l r f a +a t

T T r [ l - | 3 -b s | r f l - 0 -b t |

x F T BI/3 +b s,/3 +b t) "TTBIl-a -a s , l - a -a t (22.11) "PJ r [ 2 j 3 . + b . ( s + t ) |

"f7r|2-2a.-a.(s+t)j 1 <Ms+t)

T T r f a +a s l r f a +a t ]

T~T r f i - 3 -b s ] r f l - / 3 -b 1

I

m

FT

f £.+b .s-i

I

0.+b . t-i (1-u ) j

n

^. FT

-a -a s j

j

(1-u )

j

j

du

j=l

We note that all beta-integrals in (22.11) converge by the conditions (22.3). The following theorem is true. 193

Theorem

(22.3)-(22.4) hold

(22.12)

Let

22.2.

fix),

valid.

and

g(x) G ffl (L)

Moreover, p-n

q-m

j=l

J=l

if

the

next

the

inequality

conditions takes

place

)+sgn(y-ju) £ 0,

2sgn(c-

where q-m

p-n

q-m

H = Re

p-n

j=i

t h e n the is

next

q-m-p+n 2

j;b-[a

representation

of

j=i

the general

Laplace

convolution

(22.1)

true

,

(f*g)(x)

=

f(/3,b) ,(/3,b) m+1 q HP"n'q"m ' q P ^ ' I (a,a) ,(a,a) v

-TT ° ^ °

n+1 , p

u^Of 1 A" V l (2,3 , b ) ° ^ °

m ,.(m+n)..

J

(1-u )

J

1,n

u0 H

, l ( |(2a - l , a j 1,0X

(3 - 1

T"T Tu J!_= l

)

J

J

j

n

-l n -a -a F T v J(l-v ) j j j! = i

(22.13) x H

x H

where

sign

]

0,0 p-n,q-m

0,0 p-n,q-m

((a,a) O.b

n+l , p m+l , q

f ( a , aa )-*

|"o denotes

• |f(y

I

n+l,p

(0 b) m+l

the

, qy

r

g(y) L

composition

J= i

dv =

T T u i J!

TvjaJ

j=i

m n xj~7(l-u.)"bjf7(l-v.)aj j=i

J

j=i

of m H-transforms,

dudv

J

du = ]

[du.

j= i

||dv.. J=i

x

j=i

y

J

194

Proof. Since the conditions (22.3)-(22.5) hold then the convolution (22.1) exists

and

according

to

property

(23.14) and

representation

(20.13) we can obtain that our convolution has the next form

f(/3,b) (22.14)

p-n,q-m q»p

(f*g)(x)

,(/3,b) m + l,q

(a,a)

Th(u)l,

1 ,m

, (a,a)

n+l , p

1 ,n

where H-transform in (22.14) is defined by (16.18) and h(u) is equal to

(22.15)

h(u) =

^

F(T)U

dx ,

and

(22.16)

F(T) =

*(x-t + l/2)*(t)f (x-t + l/2)g (t)dt

2ni

for T 6 (T

Further we use the composition properties of H-transform

and by the

Theorem 16.7 and property (23.14) it can be established that h(u) is the composition chain of m+n H-transforms

in forms

(16.37),

(16.38) of a

function h (u) which is equal to

T T B||3 +b s f £ +b t ] " T T B l l - a - a s , l - a - a

h (u) = 1

(27ri) 2

\=[

{ J

J

J J \= [ I

J

J J

t|

J J J

t s (22.17) f*(s)g*(t)u

s

t

dsdt

fTrfa + as) 77rfi-^ -b s) r r r k + a t ] TT rfi-P.-bt) j=n+i^

;

j=m+i ^

J

J

Hence in accordance with condition

'j=n+i ^

J

J ;

J

j = m+i ^

J

J

(22.12) and Theorem 16.3 the next

H-transform of functions f (x) and g(x) exists in 3J1

(L)

c,7

0,0 H *p-n,q-m

n+i, p

f (s)

■[f(y)](x) = ^ j

x

ds

O.b)m+l , qs

195

j=n+iv

J

j=m+i

v

y

Finally, since each beta-function in (22.17) is bounded if the condi­ tions (22.3) take place then double integral

(22.17) converges. Using

the beta-integral

it is not difficult

(22.10) and Fubini

obtain now the representation 22.2 is proved.

theorem

(22.13) of convolution

(22.1).

to

Theorem

■

Remark 22.2. The equality (22.13) can be written in terms of the other H- or G-transforms in depending on the position of corresponding gamma-functions (see composition Theorem 16.7). Now we consider some examples of general Laplace convolution in case of G-transforms and their representations of type (22.13).

Example a

22.1. Let

in

(22.2) p = n, q = m, b. = 1,

j = l,...,m,

= 1 , j = l , . . . , n . Then we have the convolution (22.1) for G-transform

(17.1) with the kernel

*(T, = rTr(p.+x) rrr(i-«-,

(22.18)

v

j=i

y

j=i

v

The factorization equality (22.6) by the corresponding conditions in the Theorem 22.1 takes the form

„m,n n,m

(

f (a)

(a)

'■■]•[ (f*g)(u)

(0) l,m

J

L

(x)

^m,n = G

n,m

(0)

::: K f,u, i

(X)

(22.19) x Gm,n n,m

{ ^)

)

>[g(u)l(x).

1 ,n '

(0)1,m We represent the general beta-function (22.11) as follows

-A- T(20s +s+t) -A- T(2-2a -s-t)

B(s.t) = > I J 1

(22.20)

II ^ r(0.+s+t) J 1 rd-a.-s-t)

x F T B 0 +b s,0 +b t] "fTB|l-a -a s,l-a -a t|

196

Hence the analog of representation (22.13) can be constructed with the composition of m+n integro-differential operators which are defined by the Mellin correspondences 8,9 from Table 18.2 T(a+s) < TO+s)

a /3-a -(3 > x TI x

T(a-s) <

>x

i-6TS-a a-i

I

x

ro-s) Theorem 22.3. Let f(x), g(x) e !JJf (22.4) hold

valid.

convolution

(22.1) with

Then the next the kernel

m in

, 2<sp / 3 . --pp \ /

(f*g)(x) = "["Jo

X

j

j

I

(L) and the conditions

representation (22.20) is \

-p j

X

j=l V

J

j=1

i-^u. a -l i-2a j

• U. X

Laplace

true

, ja

II n

of"[o

of the general

(22.3)-

CC -1

I J

X

I

1

f m /Sj -l _

0

^ -l n

-a

-a

(l-u ) j F T v j(i-v ) j

.. FTu

(22.21)

J

J

_

J

J

0 m+n

X f

(

m 111

n 11

\

ill

/

II

) dudv

xTTCTTv. g xrid-u j^fTd-v : j=l

j=l

J

J

V j

= 1

j=i

Remark 22.3. Using the results of this section we can get many examples of convolution (22.1) and their composition representations. For classical G-transforms these examples will be considered in §24. Finally we note the associative property for convolution (22.1). For this we must define the repeated convolution (f*(g*h))(x).

Definition 22.2. Let f(x), g(x), h(x) e Til

(L). Then we shall

understand the relation (f*(g*h))(x) as the following equality (22.22)

x 1/2 (f*(g*h))(x) = (f*(x1/2(g*h)(x)))(x).

197

Supposing

the

validity

of

the

Theorem

22.1

from

factorization

property (22.6) and Definition 20.2 it follows that

(H(f*(g*h)))(x) = x" 1/2 (x 1/2 (Hf)(x)(H(g*h))(x)) = x" 1/2 (x 1/2 (Hf)(x)x" 1/2 (x 1/2 (Hg)(x)(Hh)(x))) = (Hf)(x)(Hg)(x)(Hh)(x) = x" 1/2 (x 1/2 (Hh)(x)x" 1/2 (x 1/2 (Hf)(x)

(Hg)(x)))

= (H((f*g)*h))(x).

Hence using the unity of H-transform

in space Ufi

(L) we obtain the

associative property of convolution (22.1)

(f*(g*h))(x) = ((f*g)*h)(x).

§23. G-convolution and its typical examples

The general convolution (20.3) can be simplified in the same way as in §13, when it was introduced the G-function of two variables as a (k )

special case of the H-function (10.1) with all coefficients a

(k)

,b j

j

(k = 1,2,3) equal to 1. In this section we will

consider

this simplification

(so-called

G-convolution) according to §20. The theorems of existence, representa­ tions and factorization property for the corresponding G-transforms will be established. examples

of

Moreover,

the general

it

is not difficult

convolution

to see

(20.3) contain

the

that many of elements

of

G-function of two variables (13.1). Therefore the notion of G- convolu­ tion requires independent consideration.

198

Definition 23.1. As in (13.2), for k = 1,2,3 we denote m

n k

k

(k)

]~7 ro (23.1)

* (x)

+x) J~T r(i-cx(k)-T)

j=l P

k

JfJ; °I

r V k ) + r ) f j r(i-/3
77 j=n +i k

Let also

j = m +i k

, (3

the complex numbers a j

satisfy

the following

j

!

of the poles separation supposition for the line (r = ( —-ioo,

Reipik))

+— j

— - K e ( a ( k ) ) > 0,

j = 1,2,..

j

ReUi3)

(23.2)

3

{3)

Re{a )

j = n +1, .

+ 1 > 0,

j

3

{3)

Re((B )

j = m +1, .

< 0,

j

3

type

Vs) V

1

(f»g)(x) =

the

P

3

••v

integral

\

(27ii)

Since

■ • '

o f t w o f u n c t i o n s f ( x ) , g ( x ) e !JJ1 (L) is c a l l e d t h e

next double Mellin-Barnes

(23.3)

k = 1,2;

k

j = 1 , 2 , . . . ,n ;

) < 0,

j

Then the G-convolution

—+ioo)

j = 1 , 2 , . . . , mk , k = 1 , 2 , 3

> 0,

2

2

conditions

!

functions

(s+

U f

t)

*( s ) g *

-s-t

Ct)xBtdsdt.

3

^ (x),

k = 1,2,3

are

the

kernels

of

k

corresponding G-transforms (17.1) then from Definition 17.2 we denote

*

K

k

M

k

(23.4)

c

(23.5)

•:■* Z-!"-Z»r

k

= m

k

+ n

k

j =l

-— » — 2

j =i

199

k = 1,2,3.

the

From

(20.4),

(20.5),

(20.7)

we have K = c , f i = y , k k

k

k

= 1,2,3,

and

k

P 3- C I 3 *

*

"3 = V ^

"V

Hence,in accordance with Theorem 20.1-20.3 we get the following results.

Theorem 23.1. The G-convolution g(x) € 3JT1

(23.3) exists

for

f(x) e !Jfl *

(L),

(L) if

2

2

f 1

2sgn(c +c -c ) + sgn(y +y -y ) ^ 0, & e k

k

3

k

k

k = 1,2;

3

* * * * 2sgn(c +c +c +c ) + sgn(y +y +y +^ ) > 0, * * * * * * sgn(c +c -c ) + sgn(c +c -c ) + sgn(c +c +c +c )

(23.6)

6

1

1

&

3

2

2

e

3

1

2

1

2

* * + 2sgn(y i +y 2 +* i +r 2 -y 3 ) * 0. 1/2

In this case

x

—1

(f*g)(x) <= 3J1c

defined

as

(L), where

,y 3

the

pair

(c ,y ) is

3

follows

[ (c i+ c*-c*,

VV*3}'

if

c +c < c +c ; 1 1

(c +c - c , y +y - y (23.7)

2

(c ,y ) = i 3

3

!

2

3

*

*

2

2

2

*

), 3

if

* ~

2

*

c +c < c +c ; 2

2

1 1

(c +c - c , min{y +y - y , 1

1

3

1 1 3

if

c +c = c +c . 1 1

Theorem kernels

23.2.

Let

* ( x ) , * (x) of (L), respect

and 9TI c

G-transforms the

form

ively,

i. e.

(Gf)(x)

(23.1)

exist

and on

2

(Gg)(x)

the

2

with

the

1

spaces

JIf (L) ci,yi

2'^2

(23.8)

2sgn(c +c ) + sgn(y +y ) > 0, k

Then

the

G-convolut

k

ion

k

k = 1,2.

k

(23.3) exists

for

all

f(x) e 3J1"1 c

€ W * c ,y 2 2

(L) if

and only

if

200

i

(L), g(x) l

(1

2sgn(c +c -c ) + sgn(y + y -y ) ^ 0, & k

(23.9)

k

3

k

k

3

1,2;

sgn(c +c -c ) + sgn(c +c -c ) + sgn(c +c +c +c ) 1

1

°

3 +

+ 2 s g n ( y +y 1 2

In

k

this

case

x

1/2

7

2

2

to

3

1

2

1

2

*+ * ~

7f ~V ) - 0.

1 2

3

-1

(f*g)(x) € !JJ1

c , y 3' °3

(L), where

(c , y ) is defined 3

by

3

^

(23.7). Moreoi/er, if x1/2(f*g)(x) € ffl-1 (L) for aii f(x) e ffl"1 (L) and g(x) G iJJf1 (L), then V*2

Finally,

there

G -transform

JUT1 (L) D JJJf1 (L). c '* V*3

is

the

(G f)(x) with

(23.3) by the product

following

the kernel

factorization

^ (T) for

of two G-transforms

property

the convolution

of

(f*g)(x)

(G f)(x) and (Gg)(x): 1

2

(G (f*g))(x) = (G f)(x)(G g)(x),

(23.10)

3

where the left

1

2

part of (23.10) is understood

(3)

by Definition

(3)

Theorem 23.3. Let the parameters a , /3 j (23.2). Let also the following inequalities i hold

,

20.2.

2sgn(c -c ) + sgn(y -y -1) > 0,

satisfy

the

conditions

k = 1,2;

2sgn(c +c ) + sgn(y +y -1) > 0, (23.11)

{sgn(c -c ♦

*

*

*

*

#

*

*

) + sgn(c -c ), sgn(c -c ) + sgn(c +c ) ° 1 3 2 3 1 3 1 2 sgn(c -c ) + sgn(c *+c*)i + sgn(y*+y*-y 3 -2) > 0. 2

Then the G-convolution

3

(23.3) can be represented

2

\R+ as

follows

201

by double

integral

on

(f*g)(x)

(23.12)

|_u vj

where G [x,y] is G-function

f(u)g(v)dudv uv

of two variables

of type (13.1]

* (s) * (t) 1 2 -s -t , ,. —-—? —^— x yy dsdt * (s+t)

G [x,y] = 2

(2iri)

3

(23.13)

p -n ^ 3

°

Q

M

0(3))

q -m ;m , n ;m ,n

3, 3

3

,

p

3

*3

l

1

2

;p ,q ;p ,q F M M l

l '*2

2

.0(3)) m +1 , q 3 3

x,y

2

(3)

i

(a

{3)

,t (a

^

)n

3

+1,p

:(a(1));(a(2))l 1 ,m 3

3

^

p 1 ^ q 1

) 1,

ro O n : 3

p ; ro \ (/3 q 2 {2)

(1)

);

Further we consider some typical examples of G-convolution and

their

factorization

G-transforms.

properties

(23.10)

with

the

)

(23.3)

corresponding

In fact the examples of §21 can also be related to G-

convolution. In accordance with Definition 20.2 we can affect by some G-transform on the convolution (23.3). Hence if G-transform

(Gf)(x) has

the kernel ^(x) (17.2) then from (23.3) we obtain the next equality

(23.14)

* (s) * (t) *(s+t) \T (_ 2 ^ — f*(s)g*(t)x"s-tdsdt. * (s+t)

1

(G(f*g))(x) =

{2niV

3

cr cr t s

It is not difficult to see that the last double Mellin-Barnes integral is a new convolution with the following factorization property

:(G

oG3)(f*g))(x)

(Gif)(x)(G2g)(x)

where the composition of G-transform in the left part of this equality is (by §18) also G-transform with kernel ^ ( T ) / # ( T ) . sum-convolution

(21.9)

can

be

obtained

by

the

For example, the

action

of

transform

(Kf)(x) on modified Laplace convolution (19.17). Thus this approach is the way of getting the new convolutions. 202

The most important particular case of G-function of two variables is the Kampe de Feriet function (1.7), which can be represented by (14.26) as follows

A:B;B' f (a):(b);(b'); (23.15)

-x,-y C:D;D'

(c):(d);(d');

(c),(d),(d')'

=r

(a)-s-t,(b)-s,(b*)-t,s,t

1

(a),(b)>(b>)

(2ni)2

r , I

x y dsdt (c)-s-t,(d)-s,(d*)-t

t

s l-(a): l-(b); l-(b')

(c),(d),(d')

0,A: 1,B ; 1,B*

(a),(b),(b')

A,C:B,D+1;B',D'+1

=r

x,y

Example r

23.1. Let

in

l-(c):0,l-(d);0, l-(d')

(23.15) x ,J y > 0 ,

L. = o* , L t

t

s

=
s

and

we

consider the G-convolution (23.3) with the next kernels ^ ( T ) , k = 1,2,3 k

B T,(b)-T

(23.16)

*

(T)

1

r(T)"TT r(b j -x)

U T~7 r(d U j -x)

=r (d)-x

B' "T^b'i-T

(23.17)

vl/

2

(T) U J

=r

= (d')-x

T1T(C -T)

(C)-T

(23.18)

J=l

* (T) 3

D'

(a)-x T(a - T ) j=l

203

j

B,

(rn q

= D+l,

a

q

= 1-b , j

j

2, .

= D'+l,

q

j = 1,

.,B';

,D+1;

(2)

0

in

Further

n

B' , n

*i

(1)

= A,

1-c

with

(23.4)

3

(OA)/2 and y , k

>

j = 1,...,B; j

1-a

, j-i

j = 1,...,A).

c* = — + (B-D)/2,

1,2,3, y

P.

/3 U ) = 1-d

,D'+1;

1

(B'-D')/2, c

B>

*2

j = 1,

j

j = 2,

accordance

= B, p

1-b.,

1

a(

1-d;^.

= C, p

3

2

c* = — +

2

2

2

are defined as in Theorems

k

23.1-23.3. Hence we can establish the corresponding results for the con­ volution defined below in (23.20). The factorization property and real form (23.12) for

(23.10)

this G-convolution are great interest.

In accordance with Definition 17.1

ci-(b))1>B (G

afHx)

= G

B;D + I

*• f

(23.19)

ff(u)j(x),

(i (d))

-

1>D+1

(l-(b'))

(G 2 f)(x) - G J ; J ; : + 1

1, B

1, (l-(d'))

o|"f(u)l(x)

,D'+1'

(l-(c)) (G

3f)(x)

= G

?!A

1,c 1, (l-(a))

|~f(u)~|(x)

1 , A

and by the conditions of Theorems 23.2, 23.3 the equality (23.10) holds true and G-convolution with Kampe de Feriet function

(1.7) takes the

next form of real representation (23.12)

(a),(b),(b')l r ' (23.20)

(f*g)(x) = T

(c),(d),(d')J J »

A:B;B' '

(a):(b);(b');

C:D;D'

(c):(d);(d');

F

dudv x f(u)g(v)

204

X

u'

. y V

Finally, we use the Table 4. 1 of the integral representations of Horn's

list

functions

for

the

construction

of

the

series

of

G-

convolutions in form (23.12) and their factorization properties of type (23.10). The corresponding G-transforms can be written according to the Tables

17.1

and

18.2. We

suppose

that

conditions

of

the

Theorems

23.1—23.3 hold valid and we give 11 out of 15 examples of convolutions with Horn's list functions in kernels since the corresponding 11 double Mellin-Barnes variables

integrals

(23.13).

have

the

structure

of

G-functions

of

two

The other 4 examples of Horn's functions will be

considered in §25.

Example 23.2. Let f(x),g(x) e JJl"1 (L). Then the next table of real c, d

representations (23.12) with convergence conditions (23.11) and factori­ zation properties (23.10), with existence conditions (17.5) of Gtransforms (G f)(x), k = 1,2 in 9J1 (L) for G-convolutions with Horn's k

c, d

list functions in kernels takes place. We also assume that the para­ meters of this functions satisfy the conditions

(23.2) of the poles

separation supposition (in each line of the table).

TABLE 23.1

00

" a,/3,/T" i.

00

*

r] F

(f*g)(x) = r

*

v

,

x

x

r oo> - r ^ r Adudv y;- u -,- v -^ x)f(u)g(v) l (<x,£,£ '' uv

(x1-aicx a r x r - i ) ( ( f # g ) ( x ) ) = jr(/3)(i+x)"/3|(f(x))jrO')(i+x)"/3,|(g(x )).

205

00

2.

(f«g)(x) °

= T

,

CO

F (a,|3,0;y.y' ; - - , - -

2

U V

L * > * JJ J 0

)f(u)g(v)^^-, UV

0

(( xx W = ({r[ W )) (( (( ff .. gg )) (( x x )) )) = r [ *J ]] ii F F ii o;r;-x)}(f(x)) (p:y;-x)}(f(x)) x{r[;:]iFiO';y»;-x)}(g(x)).

00 00

3.

(f*g)(x)

CO 00

= T r

F 3 ( a , a ; ^/ 3, ^^ |;y^;;-- - , - -

L y

)f(u)g(v)-^— ,

JJ J 0

0

)

A _' K x *t f( (i fg* lg K ) ) (xx)) ( xx"-*V L L 1 (l-a-|3)/2 (2X)"1 .. a e V = f| r[[^0'. aJj x Wqi -af pl =

2

^a-g '

,lA(f-r o (y-/) (J f( f( (xx )) )

2

L r , ,1 a-aij3'>/2 (2x) _1 W , 0 , , , (i)l(g(x)). x -j i op , a x e l - a - p a-/30 x J

CO 0 0

4.

( ff**gg) ()x( )x )== rr

I" a,/3

CO

1r r

[ *,?'JJ J 0

(x-(a+/3)/2LK

f_2_||

11

4

V

11V

0

x

, (l ( 1 -2r)/2 -2T)/2 f . = (x j J

xx {{

F ( aa ,, 0p ,t *r ,, *v '' ;; -- -- ,, -- -- ) f ( u ) g ( v P ^ , F

JJ yrl _i a

((aa++/3)/ 0 ) / 22) ( ( f # g ) (x)

}

,_ ,_ ^ , 1 ( y - i )) // 22 ww ,, .. ,, . w d( i--, y* ,' ) / 2 (2VxH x )(f(x))(x

(( 22 y^ x) )} lxx( (^7 -, -11) ,/ /22))(( gg (( xx )) )) ,, Reiv') [[Reir), R e ( y ) , Re(y')

206

Reij+v') ^> 1, R e(r+r')

> 3].

00

00

[r aa,/3l . p l r rr 5.

(f*g)(x) = T r

$ (a,3,y;- - , - -

& &

LL

0

U U

1

yy JJ JJ JJ 0 0

V V

)f(u)g(vP^ & &

,

UV uv

0

( x i - a i a«-yxy-i)((f.g)(x) -yxy-i)((f#g)(x)) = | r ( / 3 ) ( 1 + x ) ^ 3| |(f(x))(A ( f (x) ) ( A + g) (x) . (xi-« I ) = |ro)(i+x)-' + g)(x).

00

6.

( f *« g« ) ( x ) = r |

^

00

[ I* [* (3 3 ' y 2

L y* y

'

- - V

)f(u)H « (( vv ) ^ ^ S

U V '

JJ J 0

0

y

g ) x( x)))) ( xx-- A W_Hx (Hf (. fg» M

00

7 7..

U

{ rr (O/ 3) )( (l l++xx )) - p/ 3}| ( f ( xx )))){W r ( )^()l( +l +x x) )- -p/ '3 }' l( (gg((xx )) ) .

=

00

I" 3 1 f f (f*g)(x)

= r

L y JJ J 0

<S> ( 3 , y ; - - ,, -- -- ) f ( u ) g ( v p H ™ l , 33 U u V v UV uv

0 p = j{ rr O ( P) )( (l l+ +xx) )- - p}}( (ff((xx))))((AA+t g ) ( x ) ,

( xx -VV)y (>(
[Re(y) > 3 / 2 ] .

8.

,P

((f.g)(x) f . g ) ( x ) -^ r [f ""M lf f

d [ * v ^ ^ , , ^^. ; ;-- * *-,-*- , - * - )nu)g(v) ) f ( u ) g ( v ) d -^ - ^ ,, 1

L r.r'JJ J 0

0

(xVx«)((f.g)(x)) = {r[^]iF i O;r ; -x)}(f(x))(x (1 - 3r ' ,/2 (xVx«)((f.g)(x)) = {r[^]iF i O;r ; -x)}(f(x))(x (1 - 3r ' ,/2 \

y'-i

J

\

y'-i

J

207

9.

> x x ._, . , .dudv *T ((a,y,y ;- -,- - )f(u)g(v)

(f«g)(x) = r

2

y,y

U

(x" a A _ 1 x a )((f*g)(x)) = ( x

(1 y)/2

"

V

UV

{j y . 1 C2^)}x (y " 1)/2 )(f(x ))

(x (1 -^ )/2 {j r ,_ i (2^)}x ( ^- 1)/2 )(g(x)] [Re(y), /te(y') £ 1, /teCy+y') > 3 ] .

a, a \ fil

10.

r

(f*gHx) = r

* JJ0 y

J c

(x~*A x ) ( ( f * g ) ( x ) ) ==

x - f > r> x ,_, . , .dudv i (a,a ^,3r;- -,- - )f(u)g(v) l u v uv

{( r [ i ^ , ^ ] e - W ; ^ (I,}(f(x )) 2

jr(a , )(l+x)" a '|(g(x)).

a,/: ii

(f»g)(x) = r

" r o

X

x ._, . , .dudv

■=. (a,^,y;- -,- - )f(u)g(v)

y

2

U

V

UV

( x " V r H ( f g ) ( x ) ) = | r [ i ^ , i ^ ] e ( 2 x ) W ^ _ a (I)|(f(x))

x (A g)(x),

208

[tfe(y)>5/2],

§24. Convolutions for some classical integral transforms

As we noted above (see introduction to Chapter IV) there are some known integral transforms except Laplace, Fourier and Mel1 in transforms which

have

the

corresponding

convolutions.

These

convolutions

were

obtained in various ways and were investigated by many mathematicians.

In this section we will consider the convolutions for some classical particular

cases

of

Riemann-Liouville

G-transforms

integro-differential

transform,

the

modified

transform,

the

modified

corresponding

(see

Stieltjes sin-

and

Table

17.1)

operators, transform,

such

that

the modified the

cosine- Fourier

modified

the

Hankel Meijer

transforms.

The

results are based on the theorems in § 2 2 — § 2 3 .

But

except our approach to convolution constructions in the form of double Mellin-Barnes integrals we will use the classical way of finding the corresponding

formula

for

the

product

of

kernels

of

the

integral

transforms.

1. Case of Riemann-Liouville integro-differential operators.

Now

we

consider

the

modified Riemann-Liouville integro-differen­

tial operator which is defined by Table 17.1 (line 6) in the form of G-transform (17.1) a (x' I

+

x

) (f (x) ) s G. .

|f(u)ll(x)

1,1

(24.1) ,a-8-i

|3 f (x-u) a _ / 3 _ 1 -a_ r . . =x Jo r(a-3) u f(u)duHence in accordance with Definitions 17.1, 23.1 and Mellin corresponr(l-a - T ) dence 9 from Table 18.2 we set ^ (x) = =r^—^ r, k r11-p -T J k

209

k = 1,2,3, where

(24.2) (24.2)

[ (

Re(a Re(a ) < 1/2, 1/2,

\J

Re(a )) << 0, 0, Re(a

[[

Re(0 ReO 3)) < 0.

kk == 1,2; 1,2;

Then the the G-convolution G-convolution (23.3) (23.3) of two two functions f f(x), ( x ) , g(x) g(x) e e ffl' Ml11 (L) (L) for for our case takes the form

i(a,/3)

r r rd-i53-s-t) ru-03-s-t) ni-^-s) ra-a2-t)

(f ( f g)(x) gHX) =

u^i? J J ra-a^-s-t) r(i^-s) rd-^-t) i^77

**

cr cr X. t

(24.3)

Ss

x f*(s)g*(t)x"s'tdsdt.

Since by the formulas (23.4),

K

= K y

y 3

= Re(a-fi),

(23.5) c* = 0 , / K

K

k = 1,2,3, K

from Theorems Theorems 23.1 23.1 - 23.3 23.3 we obtain the the following following correspon-

3

ding result.

Theorem 24.1. The convoluoion 3JT1 (L) c,fr

(24.3) exisss

for

f(x), g(x) e

if

(24.4)

{

2 s g n ( c ) + sgn(y+tfe(a - 0 - a +0 )) > 0,

\J

2 s g n ( c ) + sgn(2y+tfe(a +a - 0 - 0 )) > 0,

I

2 s g n ( c ) + s g n ( 2 y + R e ( a +a - 0 - 0 - a +0 )) > 0.

v v

1

2

1 ' 2

3

k = 1,2;

3

(a,13) In this +0 ) ,

cc*

Let Let (x

2

I

x

(f*g)(x)

y+tfe(a - 0 - a +0 ) ,

*J O

ft fR

case

1/2

tt t

2 t2

C t.

O O

O O

Riemann-LioaviiJe K ie^ann-Liouviiie 22

__CCXX

x

22

)(g(x)) exist

€ 3JI"1 * ( L ) , where c,y

j

= min{y+/te(a - 0 - a 1i

2y+/te(a +a - 0 - 0 - a +6 ) ) . 1

& &

1

ttrraannssffoorrmm on the

spaces 210

^

O O

((24.1) 24.1) -1 -1

O O

__

3

-.

((xx * V I1 1

!D1 (L), c,9T c,y

1i

i.e.

')f (f '\ x ^ H C x(x)) ))

and

(24.5)

2sgn(c) + sgn(v+Re(*

-$ )) £ 0, k

Then the convolution and only

(24.3) exists

for

k = 1,2.

k

all

f(x), g(x) € 3Jf

(L) if

if

[

2sgn(c) + sgn(3r+/te(ak-£k-a3+/3 )) £ 0,

k = 1,2;

(24.6) 2 s g n ( c ) + s g n ( 2 ? + K e ( a +a -/3 -/3 - a +/3 ) ) £ 0 .

Moreover,

there

is

/3

a -/3

(x

3

I

3

the following

3

property

T(a,0)

-a x

factorization

3

)((f*g)(x))

(24.7) £ a -/3 - a |3 a - 0 - a = (x *I * *x 1 ) ( f ( x ) ) ( x 2 I 2 2 x 2 ) ( g ( x ) ) . Finally,

if

the following

inequalities

hold

ReU -0 -a +0 ) -1 > 0,

k = 1,2;

(24.8) Reio. +a -0 -0 ) -1 > max{0, Re (a -0 ) +1}. 1 2

Then

the

convolution

3

1 2

(24.3) can be represented

3

by double

integral

on

2

IR+ as

follows

(24.9)

where

«£<*> = {} G[n # ^ > ^

G [x,y] is the next

(24.10)

G-function

of two

0,1;0,1;0,1 G [x,y] = G 1, 1; 1,1; 1,1 x,y

variables

0 , 0 :a ;a 3

1 2

a :0 ;0 3

1

2

More interesting situation is represented in the case with the same kernels iels for convolution (24.3) when a = a = a = a, 0 1

2

3

Besides the Theorem 24.1 we will use Theorem 22.2. 211

= 0

= 0

= 0.

Corollary 24.1. Let Re(oc), Re{fS) < 0; a = ^ 0^ = QL^ Re(ct), Re{(3) ag ==a 30^; ; 00==01^ = 0 =fi. The convolution 2

(24.3) exists

for f ( x ) , g(x) e € 9JT1 (L),

3

if if

c,3f

sgn(3-+tfe(a-0)) ^ 0, ( 2sgn(c) + sgnij+Re(*-$)) \ [ 2sgn(c) ++ sgn(2*+fle(a-0)) sgn(2*+fle(a-0)) ** 0. 0. [ 2sgn(c)

((24.11) 24.11)

jCa.0) ^a.0) and x 11 // 22 (f*g)(x) (f*g)(x) 6 5J1"11 and e 5JT

„ R

L)).. ((L

miinn cc ,, m 2 ^ + R e (( a a -- 0 0 )) >

Moreover, there

is the following

facoorization

propprty

jCa.0) (x/3Ia"/3x-a)((f*g)(x)) = (x/3la-/3x-a)(f(x))(x^la-/3x-a)(g(x)).

(24.12)

+

+

+

Proof. From the Theorem 24.1 and the Definition

15.2 it is not

difficult to show that the conditions (24.4) are equivalent to (24.11). Hence the condition a

(24.5) for the Riemann-Liouville transform

(24.1)

a

(x^I ~^x~ ) of the functions f(x) and g(x) holds valid and we obtain the corresponding factorization property (24.12). Corollary 24.1 is proved. ■

The real representation (24.9) does not exist in this case since the conditions (24.8) are false but in accordance with Theorem 22.2 we can establish

the corresponding

representation

(22.13)

for convolution

(24.3) in case of the same parameters. In our case the Theorem 22.2 is reduced to the following form.

Theorem 24.2. Let conditions

(24.11) hold

f(x), g(x) € UK" UJT1 (L), Jte(a), Re(oc), /te(0) Re(0) < 0 and valid.

Moreover,

if

the next

inequalyty

the takes

ace

frS J.

(24.13) then the next j = 1,2,3) 1,2,3J) is

2sgn(c- 1/2) + sgn(y//te(|3)) ^ 0, representation

of the convoluoion

true 212

(24.3) (a. = a, 0. = 0,

r~ w ^ r a /3-a - 0 . , 2a-2 A 2-2a. (( f * g ) (x) = (x TI' x ' )o(x Ax )c

u

(1-u)

(24.14) , 0-1.-1 l - 0 _ w . , 3-lA-l 1-3 W ,„ * A l x (y1 A y 'f)(xu) (y' A y ' g) (x(l-u) )du

where

the

operators

in

the

right

part

of

(24.14) are

defined

in

§17 — § 1 8 .

Similarly,we can obtain the corresponding constructions for adjacent Riemann-Liouville

operator

(x I

x

)(f(x))

(see

line

8

from

Table

18.2). We note the simplest case of convolution for this operator if in kernel

*(T) =

o+ ) we

r

set

a = 0,

0 = 1.

Hence

from

(22.1)

and

Definition 15.2 the above-mentioned convolution can be represented as follows l i

(24.15)

(f*g)(x) = (lV 1 )(f(x))g(x) + (lV 1 )(g(x))f(x)

and its factorization equality takes the next form 1

(24.16)

1

l i r

(I x" )((f*g)(x)) = (I1x"1)(f(x))(I1x"1)(g(x):

2. Case of the modified Meijer transforms.

From the Table 17.1 (line 23) we have the modified Meijer transform as follows

(24

17)

J2Kv (2Vx)lo[f(u) I )I

= G'2,0 0,2

>|"f(u)j(x i^/2,-i^/2

00

=2

R

(2vx7u)f(u ,du

In formula (23.1) setting * (x) = r(-iV2+T)r(iV2+x) for all k = 1,2,3 we establish the next convolution for modified Meijer transform

213

r(^/2+s)r(-v/2+s) r(iV2+t)r(-iV2+t)

1

(f*g)(x) =

r(i>/2+s+t)r(-i>/2+s+t)

Uni)' (24.18)

f*(s)g*(t)x"s Theorem 2 4 . 3 . Let for

all

\Re(v)\

f ( x ) , g ( x ) e ffi"

1

c, y

< 1. Then

the

and only

if

convolution

(24.18)

exists

2 s g n ( c ) + s g n ( y - l ) £ 0.

(24.19)

In this

(L) if

\isdt.

case

x 1 / 2 ( f » g ) (x) e UTT1

(L).

c.min (y+l , 2^+1 )

For convolution

(24.20)

(24.18) the factorization

property

holds

J2K^(2v^)jo["(f*g)(u)l = J2K^(2v^)U Tf (u)l J2K^(2v^)|o ^ ( u ) ! .

Proof. This theorem is the corollary of the Theorem 22.1, where the conditions (22.4) are reduced to inequality (24.19) since K = 1, JI = 0, V* = - 1 .

-

For convolution (24.18) the corresponding real representation (22.8) does not exists since here conditions (22.7) are

false

and

integral

(22.9) diverges. But we can obtain the analog of the Theorem 22.3.

Theorem 24.4. Let

\Re{v)\

< 1 and f(x), g(x) € 3Jf

(L). Then

the

c, W

representation

of

the convolut

ion

(24.18) is

true l l

(f*g)(x) =

X I

X

|o|x

I

X

(u /u ) 1

2

(24.21) du du

V/2

(1-u )/(l-u ; 1

2

flA u u |g lGl(l-u 1 2J

K

214

1

1

2

)(l-u )I u u (1-u )(l-u ) • 2

J

1 2

1

2

next

3. Case of the modified Stieltjes transform.

The modified Stieltjes transform can be found

in the Table 17.1

(line 21)

l.lf

^ l

(l+x)"Pj<>rf(u)l = G***

ir{p)

(24.22)

|f(u)l(x)

0

u P f(u) du

r(p)

(x + u) p

U

It is more convenient to denote the transform (24.22) by (S f)(x). Hence P setting in (23.1) * (T) = r(x)r(a -x), k = 1,2,3 we establish the k

k

convolution for modified Stieltjes transform in the next form

r(s)r(a -s) r(t)r(a -t) (24.23) (f*g)(x)

-

f (s)g (t)x

cisdt.

r(s+t)r(a -s-t)

UniV

3 t s

In accordance with (23.4)-(23.5) we have for convolution (24.23) that c r = 1-Re(a l-Re(oc c* = 1, y* k

k

) , k = 1, 1,2,3 and y = y . From Theorems 23.1 - 23.3 ), kk

3

3

we get the following result. Theorem 24.5. Let Re(oc ) > 1/2, j = 1,2, Re(oc ) > 1. The tion

(24.23) exists

for

all

1

f(x), g(x) € JJT (L) if

and only

convolu­

if

c,y

2sgn(c) + sgn(r+/te(a - a )) £ 0,

k = 1,2;

(24.24) 2sgn(c) + sgn(2r+l+ite(a - a - a )) £ 0.

In

this

( f * g ) ( x ) e 3JT (L), c,r# j + Reioc -a ), 2y+l+Re(a -a -a )}. 3

case

2

Moreover, (24.25)

x

3

there

is

* *

= min{y+Re{oc - a ), 3 l

1 2

the following

factorization

s a (S (f»g)(x)) = (S f(x))(S 3

where

1

g(x)). 2

215

property

Finally,

if

the

following

inequalities

hold

Re(a -a ) > 1, 3

(24.26) Reia Then

the

convolution

(23.12) on IR+ with

-a -a

3

l

2

) > 1.

(24.23) can be corresponding

k = 1,2;

k

represented

G-function

of

by

double

two variables

integral (23.13).

Now we consider the convolution (24.23) for a 3 = a +a 1 2. It is not difficult to see that the last inequality of (24.26) is false in this case. Therefore, the real representation (23.12) on (R+ does not exists. Nevertheless, we have the corresponding relation as in (22.20).

Theorem 24.6. Let € TF1 (L). Then c,r true

(24.27)

the

Reict

) > 1/2, j = 1,2, a j

next

representation

of

a -l a -l ,/ u u 1 (1-u ) 2 f

(f*g)(x)

the

convolution

, 1-u

>.

This formula

can be

x

obtained

(22.10), Definition 15.2 and Fubini theorem.

j

\

by

and f(x), g(x)

1 2

(24.23)

^

i A A-T=£-

\

Proof.

= a +a 3

\

using

is

du du

x

(1-u2

u >

\

the

i

beta-integral

■

If, instead of convolution (24.23), we consider the next integral

(T(s)r(a -s) r(t)r(a -t) (24.28) (f*g)(x) = (2ni ]

f*(s)g*(t)x"s_tdsdt, r(-l/2+s+t)r(a +l/2-s-t) 3

(r cr t s

where Reict

) > 1/2, then we can also obtain the theorem of existence and 3

factorization property in form

(24.29)

x" 1 / 2 (S

a

(y1/2(f*g)(y)))(x) = (S

a f)(x)(S a g)(x). °

°

3

1

216

2

Moreover, we can evaluate the corresponding G-function of two variables (23.13) by means of double hypergeometric type series (see (8.18)).

Let

the

kernel

of

the

convolution

(24.28) be

#(s,t)

and

the

corresponding G-function of two variables G [x,y] be defined by means of formula

(23.13). The integral


Re{f3),

Reir)}

(23.13) for G

[x,y] converges when min

> 1/2, x,y € IR+ and the conditions

(24.26) are

true.

The purpose of this point is to find the representation of G [x,y] by means of Horn's list of functions

(see §4). The method which we use

contains the repeated applications of the Slater theorem

(0.I.Marichev

(1983)) for getting the value of Mellin-Barnes integral from the sum of the residues in the left (or right) poles. For this we will subdivide a quarter of the plane R

into four domains: { 0 < x , y < l > ,

{ 0 < x <

1 <

y>, {0 < y < 1 < x >,{1 < x, 1 < y}.

1. Case

0 < x, y < 1.

As Re{oc ), Re{oc ) > 1/2 then the integral of 1

2

the type (23.13) with the kernel & (s,t) can be evaluated repeatedly from the sum of the residues in the left poles of the gamma-functions at first for s = -m and then t = -n. Finally, we get the next representa­ tion

*'*.*-£ ^

A" r a

a +m,

a +n

1

2

+ — +m+n, - — -m-n 3

2

2

(24.30) a ,a 1 27T

2.

1

,a + — L

where F

3

3

1

F ( — , a , a ; — + a ; x , yJ ) ,

2

2

1

2

2

3

2

is an Appell series (A.Erdelyi et al. (1953)).

Case

0 < x < 1 < y.

In

this

two-dimensional

Mellin-Barnes

integral of the type (23.13) can be evaluated from the sum of residues in the left poles s = -m and the right poles t = a +n and we will have

217

i

/-r

\

G [ x , yJ ] = )

(-1 J

m+n

. a +n m,1. 2

l_^ m!n!

a +m, a +n, 1

_

—;—:— x ( - )

r

y

2 l

l

a + — +m-a - n , - — -m+a +n 3

2

2

2

2

m , n=0

(24.31) a , a

=r

1

1

2

,

3

2

2

1

3

G (a , a , — + a - a , — - a 2

— +a - a , a - — 2

where G

2^

yJ

1

1

2

2

2

3

2

1 . y

; - x , - — J,

2

2

is a Horn series (A.Erdelyi et al. (1953)).

3. Case 0 < y < 1 < x.

Here the next result is true

a ,a G [x,y] = r

1

1

2

1

— +a - a , a - — 2

3

1

1

2

(24.32)

x

G (a , a , — + a - a , — - a 2

4. Case

1 < x,

1 < y.

1 2 2

=

;- —,-y). 1

X

The following result takes place

a G [x,y]

1 3 2

-•

,a 1

2K1

2

?+-

(24.33) -oc

-<x 1

xx

2

y

. „ r- (

3

.

a

.

,

1

^

.,

1

^

F ( — +cx +a , a , a ; — +a +a +a ; — , - ) . 1

1 2

2

1

2

2

1

2

3

x

y

The value of the function G[x,y] at x = 1 (or y = 1) is defined by the continuity of G[x,y] and can be obtained from Gaussian summation theorem (8.20).

For the case of the same kernels (a = a, j = 1,2,3) of convolution j

(24.23) as it is not difficult to see that conditions (22.7) are false. Hence the real representation (22.8) does not exist and we can apply the Theorem 22.3 for getting the relation (22.20).

Now we distinguish a more interesting case of convolution for

ordinary ordir

Stieltjes

transform

(24.22)

(p = 1),

when

(24.23)

a

= 1 j

j = 1,2,3.

218

Hence, using the addition formula for gamma-function we can obtain the next representation

sin(n(s+t))

sin(7is)cos(7rt )+cos(7rs)sin(7it)

r(s+t)r(l-s-t)

r(s)r(i-s)r(i/2+t)r(i/2-t)

r(t)r(i-t)r(i/2+s)ru/2-sr

Then substituting this identity in (24.23) and applying Definition 15.2, s the convolution (24.23) (f*g)(x) for a. = 1, j = 1,2,3 brings to the following one

(f*g)(x) = 7T f(x)

2712

r(t)ra-t) • . -t,. € r(i/2+t)r(i/2-t).g (t)x dt

(24.34)

r(s)rd-s) * . -s, 2rTi r(i/2+s)r(i/2-s).f (s)x ds

+ g(x)

Further, since

O.I.Marichev

r(s)rd-s) r(i/2+s)rd/2-s)

ctgdrs)

and the next

integral

(see

(1983))

s-l

(24.35)

exists

1-U

in principal

du = 7i c t g U s ) ,

value

in point

0 < fie(s) < 1,

u = 1 we

can prove

the

following

Parceval equality of type (15.4) in space !IJ1 (L)

(24.36)

2TT

i

r(s)rd-s) * , -s r(i/2+s)rd/2-s)1 lsJX as

219

=_^

n

f(t) t-x

dt = (Sf)(x), x > 0,

where (SO( x ) is

a

singular operator on half-axis (0, +00) (see S.G.Samko,

A.A.Kilbas and O.I.Marichev (1987)).

This fact

can be established

by representing

the right

part

of

equality (24.36) as follows x-l/P

f(t)

(24.37)

t-x

dt =

N

f(t)

lim

t-x

M, N, P—X» 1/M

dt.

x+l/P

Hence applying the Definition 15.2, changing the order of integration and using the uniform boundedness of inside integrals by the aid of Lebesgue theorem we establish (24.36).

Thus the convolution for Stieltjes transform (S f)(x) takes the next form

(24.38)

(f»g)( x)

= Trjf (x)(Sg)(x) + g(x)(SO(

.,].

The equality (24.36) allows to solve the question of evaluating the norm of singular integral operator(24.36) in 3R~ (L).

Actually, from (15.6) and the inequality |ctg(7rs)| < 1, s € (r we have +00

(24.39)

||Sf| _,

= ||ctg(Ti(^-+it))||f*(^r+it)dt|

im'V)

|f*(-L+it)|dt

l f l -x ' m (D

Hence |S| ^ 1. But if we consider the next sequence of functions JR (U f (x) € im'^L), n = 1,2,..., where

220

t f *(

r i . • J. \ %» +lt))

JL+it) = <| COS(TI( 2

t ^ -n,

2

t > -n, then it is not difficult to obtain that for any e > 0 there exists such number n, that

n

Thus US!

m a(D

n

1

m

(D

=1.

4.Case of the modified Hankel transform.

The modified Hankel transform can be defined by line 9 of Table 17.1 and line 13 of the Table 18.2 as follows

f (a-/3+i)/2. (n/-\\fr( ^ _ ~1,0 J |x a+P-i(2^)}(f(x)) =G0,2 (24.40)

>|f(u)l(x) [ a,l-0J

00

_

"

x
JVi (2vx/u) u

f(u)—

Then the G-convolution (23.3) of two functions f(x), g(x) € OTf (L) for our case takes the form

r rifi -s-t) r(a +s) r(a +t) (24.41)

(f*g)(x)

' 2

(2iri)

3

1

2

r(a +s+t) r o -s) r o -t)
t s

x f*(s)g*(t)x"s_tdsdt. For

this convolution all

transforms approach

of

can

easily

obtaining

results just as

be the

obtained.

Now

convolution

the Hankel transform (24.40), when a

221

in the previous we

for

= 0, 1-0

consider one

the

particular

= -v,

cases of classical case

of

j = 1,2,3. Hence,

if Re{v)

^ 0 from a stronger

O.I.Marichev

theorem

and S.B.Yakubovich

than 17.2

(see Vu Kim Tuan,

(1986)) it follows

that

the Hankel

transform can be represented in the form

(24.42)

f J / ) ( x ) = |x" y/2 J^(2v^)|(f (x ))

J (2/~^7u")u

f(u)du,

v

x > 0.

In accordance with the Theorem

17.1 if f(x), g(x) € UJ1 (L), then

|J f|(x), (j g](x) € S J T ^ ^ ^ a )

and

0, Re

(V)

by

Consequence

15.2

h(x) = x 1 / 2

(j fl(x)fj gl(x) e 3TT1 , (L). IV J { V J 0, Re (V) The inverse transform to

J f (x) denote by

to the Theorem 16.4 for f(x) € 3H

(x) and according

(L) -v,0

f (x) = G0,1 2,0

W

(24.43)

J f

)

>[f(u)j(x)

For our future convolution it is natural to require the fulfillment of

the

factorization

property

of

type

(22.6).

In accordance J

Definition 20.2 we represent the convolution

x- 1/2 [3^h)(x) c),

where

the

operator

with

v

(f*g)(x) sought for as

fj^fl(x) fo r

f(x) e

ffl

(L)

0,Re(y)

(similar to (24.42) for 0 < Reiv)

(24.44)

M"i

(x) = X

^ 1) has the form

J (2/u7x)u '*

w

"f(u)du,

x > 0.

Thus, we have the next repeated integral for convolution (f*g)(x) 222

(f*g)(x) = X

r-\(

T M/

,

.-V/2.-1/2

1/2

J (2v u/x)(x/u)

u

J (2VHLI7V)(U/V)

(24.45)

,-v/2 , ,dw " g(w) — du — . x f(v)dv JV (2vu7w)(u/w) ° w u

—1

Theorem —

24.7.

Let

f(x),

1^/p-^/A

g(x) € Jli (L)flL(x

< Re(y) 5 1 Then x1/2(f*g)(x) e !JJ1 * (L) and the following

;R ).

and

represen­

tation is true

(f*g)(x)

r(,)x-2p [ fr l-(Vx/u 2TT r(2v) J J L

- Vx/v)"

D(x,u,v) (24.46) (vxTu + vxTv) -1

where the domain

(24.47)

Proof.

D(x,u,v) is defined

/x/u - vx/v

u

v

by

the

f(u)g(v)dudv,

next

x > 0,

inequality

< 1 < vx/u + vx/v

In accordance with the conditions of the theorem and

asymptotic behaviour of Bessel function it is not difficult to shown that the repeated integral (24.45) absolutely converges. Hence by Fubini theorem we change the order of integration and the inside integral is the known Sonine integral

(see formulae 2.12. and 41.14, Vol.2 from

A.P.Prudnikov et al. (1986))

223

V-l/2 V-l/2 (

0 l-3l'

r

?

2 -(a-b)22 cc2 -(a-b)

-i J

VH abc)"r(i>+l/2)L V^ ((abc)^r(i^+l/2)L 00 00

I

J

1^-1/2

r

v-l/2

V 1_y

u~ u JJ \

V V

(au)J (bu)J (cu)du == J V V

xx [(a+b)22-c22l , .

V V

if |a-b|
J

|_

a,b,c > 0, Reiv)

o 0 (

Thus by

the simple

change

0, 0,

> -1/2,

in in other other cases. cases.

of variables

in

integral

(24.45) and by

the evaluation of Sonine integral we may come to representation (24.46). j

v

The statement x1/2(f*g)(x) s 3T-1 (L) and The statement x1/2(f*g)(x) e Jj-1 (L) and Thoperty Theperty

corresponding corresponding

factorization factorization

(vi),,,. (V]<«>M<"

(24.48)

can easily be obtained by Theorem 17.1 and Consequence 15.2. Finally,

we

try

to

distribute

the

convolution

■

(24.46)

on

the

modified Sin-transform

00 (X)

G

This

n,o o[f(u)](x) = — i - sin(2v^Al)u"1/2f(u)du, 0> 0> 2[0,-1/2j 2[0,-1/2j L L -I I vv^xj ^J o

transform

can

be

obtained

from

(24.40),

(v = 1/2). But in the Theorem 24.7 Re{v)

when

x > 0.

a = 0,

0 = 3/2

> 1/2 and we will consider the J

corresponding convolution as

(f*g)(x) = lim (f*g)(x). From representa1/

tion (24.46) we get the next convolution 224

7 1 / tL +

Si n

(24.49)

-1/2

(f*g)(x)

u

2VTC x

-l/2_,

v

^

^,

f

■,

„

f(u)g(v)dudv,

~

x > 0,

D(x,u,v)

where the domain D(x,u,v) is defined by the inequality (24.47).

The convolution for cosine Fourier transform is easily obtained by simple

trigonometric

formula

for

product

of

two

cosines

(see

also

I.N.Sneddon (1951)).

§25. Modified H-convolution

The idea of this section is based on Remark 15.2 where we have noted that it can be established a more general property of belonging to space 5H

(L) of the product of two functions. Similar

to the problem of

c , <J

function product convolution

in these

object.

Hence

spaces we

can

it

difficult

is not

consider to

the

corresponding

bring

the

double

integral (20.3) to the next modification of H-convolution

& (s) $ (t) 1 2 * . * . -As-Wt, ,, , ,. _,_ . , f (s)g (t)x dsdt, $ (As+wt) °

(f*g)„ Cx) = ,_1 .,2 (A,w)

(25.1)

(2711 )

3 cr cr t s

where $ ( T ) , k = 1,2,3 are defined by (20.1) and A,w € (R, Aw * 0. This k

modification

is very useful

for obtaining

some new convolutions and

their factorization equalities which will be considered below. According to conditions

(20.2) the corresponding

inequalities for

parameters of the modified convolution (25.1) will be written as follows 225

k t( Re(^ )) k)) tfe(/3<

I I

(25.2)

1 I I

! k l) >>0, 0, ^i bb |

j = 1,2 m k, , j 1,2,...,m k

k = 1,2,3; k

k> lk) 1-Re(«! )--La! k) > 0, 0, l-Re{alk))-±-a j 2 j j j 2

j 1,2,...,n j = 1,2 n , , k k

k = 1,2; 1,2; k

++

, „ , (3). (3>A+U 3

1l _-

„.

R e t a )^_ W ' ^

Re ( a j

>>

aj - r

0,

°-

Re(a<3>) > 0, 0> Re(a'3)) + + a a !"3>) ^ ^> J J *J J *i3) (3)

( 3 )

j

j

I l-^e(/3 l-Re{(B )) - b

j j=1 1,2,.,.,n ,2 n,, , 3

j p , j = = n +1 +l,...,p 3 3

3 3

^ > 0, 0, jj = m 33 + ll

qqg3.

2

Similarly to asymptotic estimation (20.6) for the kernel in convolution (25.1) we have $ (s) $ (t) VAs^t) "Se*P

, >n [-(sIs.tl-Kjsl-^ltlJj

r

-hxi^n *Eexp Ksi-ti-Kjsi-^itiJj 3

x |s+t| 3 | s | 3

a a

x |s+t| IsI

It1

2

It1

2

, ,

where the ordered pairs U ,/i ), k = 1,2,3 are defined by (20.4), (20.5) k k and the value JI is evaluated follows where the ordered pairs U ,/i as ), k = 1,2,3 are defined by (20.4), (20.5) k k and the value JI is evaluated as follows 3

{3)

~* = ji* = M 3++^|li ^ " M^ £ 3a(3) a(3) _ - E 3 bl\( 3 )} . [j=i J j=i J J

(25.3)

Now

we formulate

the corresponding

representations for modified H-convolution

theorems

of existence and

(25.1) which take the forms

of the Theorems 20.1-20.3 in case X = w = 1 and they can be proved in the same way but only with the help of the corresponding estimations in §20.

Theorem 25.1. The convolution

g(x) € 3JT1

c ,y 2 2 2 2

(25.1) exists

(L) if

226

1 for f(x) €ffi ffi" * (L), (L), c ,y c1 .y, 1

C +K

2sgn( l^,1 - K 3 ) + sgnCy^ f^-J^) * 0,

f

C +K

2sgn( ? ,2 - K 3 ) + sgn(r2+ ^"M*) * 0, C +K

(25.4)

C +K

2sgn(-|^p + - 4 ^ 4 ) + sgn(ri+ * 2 + jn^ JLX^ ) ^ 0, C +K / 1 1

sgn(

_K )

^r

C +K , 2 2

x ,

sgn(

3

C +K r 1 1

v

C +K 2 2.

T^r " K3} + s g n ( ^q- + -ra- }

+ 2sgn(? + y + M + l^~M ) - °-

In this case x (c , y ) is defined 3

(f*g) .

as

(L), where

(x) <= 5J1

(A, w)

c ,3r 3

the

pair

^

3

follows

3 r

C +K.

(

^ (Cl+Kl)/lAl < ( C 2 + K 2 ) / I W I

-|AT" V W V '

C +K

( f ,2 3

3

K

U

(25.5) (c ,r )

3

, y +ji -M*), 2 *2

i f (c

3

+K

2

2

Theorem 25.2. Let H-transforms

lT (C +#C }/ X

1

1 I I

-l

2

$ (x),

2

$ (x)

(20.1)

(L), respectively,

3J1

)/|A| 1

l

i.e.

exist

=

1

(C +K )/ W

2 2 I I

(Hf)(x) and (Hg)(x) with 1

1

+K

l

'

'

C +K

VWVV 3 '

kernels

) / | w | < (c

on

the

the

2

spaces

the conditions

O T C

c ,y l i

(20.20) hold

(L)

and

valid.

the

Then

2

convolution

25.1) exists

for all

(L), g(x) e m

f(x) e 311 1 1

and oniy if 227

L) if 2

2

C +K

^

2sgn( K . 1

- K3) + s g n ( r a + J ^ " ^ )

C +K

2sgn(

* 0,

^#

- T 3 T " K3)

+ s

e n ( V f2-»*3) * °.

(25.6) C +K , 1 1

Sgn(

v

-Txr " K 3 )

+sgn(

,

C +K 2 2

" T ^ ~K3]

. ,

+ sgn(

,

C +K 1 1 ,

C +K 2 2.

3

nrr ra^

+ 2sgn(y i +y 2 +M 1 + M 2 -M 3 ) * 0.

/n this case x ( A + W (c ,y ) is € m'1

defined

(L) for

by

1)/2

(25.5), and

f(x) e 5JT1

all

(x) e UJf1

(f*g) .

moreover,

(L), where

3

3

if

x

the pair

(f*g)

(L) and g(x) e 3J1"1

(x)

(L), then 3H"1 (L) D

(L).

mr3

3

Finally,

there

E-transform

is

the

(H f)(x) with

the

following kernel

$ (T) for

3

(25.7)

3

the

left

Remark

15.2.

convolut

ion

of (25.1)

)(x) = (H f)(xA)(H g)(x W ),

° (A,W)

part

of

Theorem 25.3. Let conditions

the

property

3

(H (f»g),.

where

factorization

(25.2). Let

1

(25.7) is

the also

understood

parameters the

2

following

228

a. ,

by

Definition

20.2 and

/3. , a. , b. satisfy

inequalities

hold

the

2sgn(ic -1X|#c ) + sgn(ji -fi - 1 ) > 0 2sgn(u , - | W | K 3 ) + sgn(/i2-ji*-l) > 0, (25.8)

2sgn(K +K ) + sgn(ii +u -±, - 1 ) -> ^, 0 1

1 2

2

2min^sgn(K njsgn(Ki-|A|jc — |A|#c3) ) ++ sgn(K sgn(ic2 "|W|K - | w | i c3 ),) , sgnCic sgn(ic -|A|K - | A | i c ) + sgn(ic +*c ) , sgn(K ) + sgn(K +K )i + sgnt/^+^-fi*K 2-|w|ic 2 ~ I W I K 33 ) + S g n ( l C l i+ I C 22 ) | + S S n ^ 1 + ^ 2 " * 1 3 - 2 ) > ° -

Then

the

IR as

convolution

represented

by double

integral

on

follows

^,M^=ff«[^>'^

(25.9)

where

(25.1) can be

H[x,y] is H-function

of two variables

(20.27).

Now we consider some examples of modified convolution (25.1) in form (25.9) and factorization property (25.7).The corresponding conditions of validity can be obtained by the Theorems 25.1 - 25.3.

Example 25.1. Let A = w = 1/2, $ (x) = $ (x) = $ (x) = T(x). Then 1

2

3

the integral (20.27) is convergent by the conditions Legendre's (21.13)

doubling

formula

for

gamma-function

(25.8) and by

and

identity

the corresponding H-function H [x,y] is equal to H[x,y]

r(s)nt) x -s y -t d s d t

= {2niY t s

M«L I ^ t i r ^ 2 (2*i)2 I I F ( s + t ) t s

2Vii

exp

mn 229

tdsdt

Hence we obtain the convolution

eXp

= ^ _

(25.10) ( f * g ) ( 1 / 2 > 1 / 2 ) ( x )

[" X H

f (u) g (v)

dudv uv

with factorization equality

(25.11)

(A (f*g) )(x) = (A f ) ( v 5 ) ( A g ) 0 / x ) . + (1/2,1/2) + +

The more general by Gauss-Legendre

example of convolution

product

formula

(25.10) can be

for gamma-function

(see

established O.I.Marichev

(1983)).

Example 25.2. $ ( T ) = T(x). r = 2,3,...

Let

Then

and

A = u = 1/r,

the

using

the

r = 2,3

integral

(20.27)

reduction

property

H-function can be represented by the following

7—x

i2ni)

r

is

* (T) = * (T) = 1 2

convergent

(13.6)

the

for

y

dsdt

r

-1/2, _ .(l-r)/2 (27i)

r(s)f(t)

r(s+t)

UniV

^.fs+t+j)

I I1 Iv r IJ (x/r)

(y/r)

j=i

t s

=r

1/2,_

. (l-r)/2 r r - l , 0 G 0 , r-1

(2TT)

m'

Hence we obtain the convolution

230

l/r, 2/r,

all

corresponding

equalities

r(s)nt) -s -t, ..

H[x,y]

(25.12)

and

(r-l)/r

dsdt

t r-

f*g&

t

A (l/r,l/r)

^

1/2,

,(l-r)/2

0 (x) = r (27 i)

(25.13)

jr-1, 0 J 0 ,r-l

"(j ^ ) r

1/r, 2/r,,

(r-l)/r

dudv f(u)g(v) uv

with factorization equality

(25.14)

)(x) = (A f)(rVx)(A g)( r Vx).

(A (f*g) +

( l/r, l / r )

+

+

It is necessary to note that the case r = 1 brings to the kernel of modified Laplace convolution (19.16), but the corresponding H-integral (20.27) diverges due to conditions

(25.8). Therefore

the convolution

(19.16) has no representation (25.9).

Example 25.3. Let A = w = — , $ (x) = $ (T) = T ( T ) and $ (T) = r

2

1

2

3

r(x)/r(i;-T). Then similarly we have the next formulas

(f*g) &

(x) (1/2,1/2)

(25.15)

hffl ]

T(y+l/2)

2Vn~

UK

J 1>-1

r-f , f ,1/u+l/v, , f (u) to g (v) dudv, uv

(2rfol x(y"1)/2)((f*g) to J

(1/2,1/2)

( x ))

(25.16) = (A f ) ( V x ) ( A

g)(i/x).

Now we return to §23,

where it is noted that in Table 4.1 there are

four examples of Horn's functions, which have the integral representa­ tions not satisfying the structure of G-convolution 231

(23.3) (see lines

5 — 6,

14 — 15).

But it is not difficult

to find that two functions

H (a,/3;y,«5;-x,-y) (line 6) and H (a; y, 8; -x, -y) 4

(line 15) satisfy

the

7

structure of modified H-convolution (25.1) when A = 2, w = 1.

Example

25.4. The

next

pairs

of

convolutions

and

factorization

equalities

a,0

{f

*z\2

(25.17)

„

(x) = r 1}

f

s

0

x

*

\r-r

\

r

^dudv

// (a,£;y,5;- -,- - )f(u)g(v) 4

(x"aA'1x<X)((f*g)/

U

&

V

UV

(x))

(2,1)

= {rg] i F i O;y;-x)}(f(x 2 )) (x ( 1 " 5 ) / 2 { J ^ t t V x ) } x (6 " 1)/2 ) (g(x)),

(f*g) (2)i) (x) = r

(25.18)

r ,8

„ , ~ x x »_, . , .dudv // (a;y,<5;- -,- - )f(u)g(v) , 7 u v uv

Re(y+<5) > 3,

(x~V 1 x a )((f*g) /

(x))

(2,1)

(x<1 r)/2

" { v/2^}*'9"1'^'^^5

* (x U - 5,/2 { J a _ i (2^)}x <3 " 1)/2 )(g(x)),

take place for some functions f(x), g(x) € !Jfi

(L), where the parameters

c, d

of space

and

of

the kernels

satisfy

25.1 — 2 5 . 3 .

232

the conditions

of

the Theorem

§26. General Leibniz rules and their integral analogs

In

this

last

section

we

consider

the

original

application

of

G-convolution (23.3) to constructing new Leibniz rules for some Mellin type transforms, namely G-transforms (17.1).

Leibniz rules for the operators of fractional calculus and their integral analogs were considered by several authors (see, for example, Y. Watanabe (1931), T.J. Osier (1971), (1972a,b), S.G.Samko,

A.A.Kilbas

and O.I.Marichev

17.1

(1987)).

operators are known

In fact, as is shown in Table

to belong

these

to the class of G-transforms. Conse­

quently, it seems very natural to try to obtain some new Leibniz rules and their integral analogs for the others G-transforms similarly as for fractional integro-differential operators, i.e. as series or integral representation of some G-transform of the product of functions f(x)g(x). At first in papers of S.B.Yakubovich and Yu.F.Luchko

(1991a,b) it has

been found a general method for constructing such Leibniz rules and their integral analogs on the basis of notion of the G-

convolution

(23.3). We will use the following formulas and summation theorems in the further discussion

(see also L.J.Slater

(1966), A.P.Prudnikov

(1986), Vol.2.):

(26.1)

T(z)rd-z) = 7T/sin(7rz)

(Addition formula for gamma-function).

Gaussian summation theorem: a,b; (26.2)

F 2 1

1

[c,c-a-b] [c-a,c-bj '

233

Ke(c-a-b) > 0.

et al.

Saalschutz*s summation theorem:

(c-a) (c-b)

-n,a,b; (26.3)

F

n

1

3 2

n

(c) (c-a-b)

c,1+a+b-n-c;

Dixon*s summation theorem:

a, b, c; (26.4)

[l+a/2,1+a-b,1+a-c,l+a/2-b-c] [l+a,l+a/2-b,l+a/2-c,1+a-b-cJ'

F 3 2

1+a-b,1+a-c;

fle(a-2b-2c)

> -2.

Dougal1's formula:

(26.5)

1 f r[ a + n ^ + n l = . . f. . h , r['[c+d-a-b-1 c-a,d-a,c-b,d-b /

|_c+n,d+nj

sin(7iajsin(7ibj

[_<

Re(a+d-c-d) < - 1 ; a,b £

(26.6)

|_a+u,b+u,c-u,d-uj

[a+c-1,a+d-1,b+c-1,b+d-lj'

fie(a+b+c+d) > 3.

c+d-2

(26.7)

(26.8)

c+u,d

] du = —T(c+d-l]

[~a+iu,b+iu,a-iu,b-iu~|

L

_

J

Re(c+d) > 1.

_

i-2a-2b 3/2 f"2a,2b,a+b] n

Reia) > 0, Reib) > 0. 234

[ a + b + 1 / 2 J'

We also introduce the following operators, which are the particular cases of G-transform (17.1) in addition to Table 17.1:

1) The operator with Tricomi function ^(a,b,x) in the kernel and its inverse: , a.b -a_. , . -2,0 (x * x f )(x) = G. ' a 1,2

fa+1+a-bl

Jf(u)j(x

-x/u_, . . . -a-i _, x , e $(a,b,x/u)u f(u)du,

( a,a+l-b * l\ '

2) The

(x

operator

Al x v

x"af)(x) =G1-°\ '

with

f)(x)

algebraic function in the kernel:

= x

M

(1+•!-x/u)

+(1-Vl-x/u)

u

f (u )du

A/1-x/u -v/uL

2Vn

3) The

o f(u) (x); L -I

a+l+a-b

operator

with

_ , a+i>/2,a+iV2+ • 20 , 0 | 2 2,2

G!

generalized

Laguerre

|"f(u)j(x);

polynomial

L (z) k

kernel:

(x L x k

f)(x)

1

G

r

l,l

fa+1,a+A+1

Tf(u)l(x)

TT 2,l

e

a+k+A+1

-u/xT A., . v -a-i_, . , L (u/x)u f(u)du; k

235

in

the

4) The inverse operator to the operator with Kummer function

F (a;b;-z)

in the kernel: , a( T(b) r b V1 - a _ w , t-TUTFa } X f H x )

(X

=G

_0,1 2,1

f oc+l-b,a

>|~f(u)l(x) a+l-a

5) The operator with Gauss hypergeometric function

F (a,b;c;x) in the

kernel: (a+l-a.a+l-b) a,b

r(a)

2,2

>|f(u)l(x) a,a+l-c

[/.

(a,b;c;-)H(l- -) + x =-, TFTTT—£T u u r(c-a)r(l+a-b)

F (a,l+a-c;l+a-b;-)H(- -1 )]u"a_1f(u)du. 2 1 X U J

Then

setting

in (23.3) ^ (s) = $ (s) = 1 and using

(26.1) - (26.8) for representations

the formulas

of the G-convolutions

kernels in

(23.3) we obtain the following theorems.

Theorem 26.1. Let f € m

(L), g € 3R

c ,7 1 1

(r for (f*g)(x) (23.3). Then the following definitions

of the operators

1. If H(x) = r(y-ct-0-T),

(L) and *

c ,y 2 2

Leibniz

3

rules

(x = H x , X € hold true

also see in Table 17.1J:

then

(xa+/3 *A x r " a ^)(f(x)g(x)) = (f*g)(x) (26.9) ^ ^

7-1 o 7-1 o , a-7. 7-a _ w w £-7. 7-$ w . (x 'L^ x° f)(x)(x' °L k x° 'g)(x)

236

(the

under

conditions: Re{?-ct-p)

> 1,

Reioc) > -— ,

Ke(|3) > - — ,

(26.10) 2sgn( C i ) + sgnCr^eCr^jpJO+l) * 0, i = J U .

2. If

H(T) = r(r+S-a-/3-l-x), (x

' A x

then

^

) ( f ( x ) g ( x ) ) = (f*g)(x)

(26.11) 1 = V (xy"V+^"axa-yf)(x)(x5"V+a-^-5 /^ r(l-a-n)r(l-^-n)lX Vp X *Mxnx *n+a x

under

*)(x) g^xj

conditions: Re(y+8-oc-p) > ±- , a,0 * Z, /te(y-a) > - ^ - , /te(r~0) > - y-

(26.12)

Jte(S-a) > - — , Re{6-&) > -— , 2sgn(c i ) + sgn(y i + Re(a+/3-2J y }o) £ 0, i = J 2 | -

3. If H ( T ) = l / T ( r - 0 - x ) ,

then

(x y ^ 1 x ^ y ) ( f ( x ) g ( x ) ) = (f*g)(x) (26.13)

= l

r J

^ ^

a

- r

}x-af)(x)(x^{Fy.a}x^g)(x)

k=0

under

conditions: Reiv-p)

(26.14)

> - \ .

^e(a) > - ■!■ ,

Ke(/3) > — , 2sgn(ci-{1f})

fle(y-a-/3)

+

> -—,

sgn(V*e{y«a}

237

+

{_? } ) , 0. i = { > } .

4. If H(x) = r(a+/3+n+x)/T(a+/3+x),

then

(-i)"(x-^mnx-^),f.x)g(x)) (26.15)

= (f*g)(x)

n

= (-l)n ye(x a (-^)Va)(f(x))(x ptn (-^) n "V P )(g(x)), = ( - 1 ) nkV= 0 C^(xa(4j)kx"a)(f(x)) (x'3+n(^Lpx-'3) (g(x)), k=0

where

k

C are binomial

coefficients

n

under

n € N,

fte(a)

fie(a+/3)

> - — , 2 s g n ( c . ) + s g n ( y . - n ) * 0, i = 1,2.

(26.16)

> -— ,

Re((S) Re(j3)

conditions:

22

> -— —, 22

(.1)^ ^ -- (( A A || nn (( xx -- ^^ ff (( XX )) gg (( xx )) ))

= (f*g)(x) (f*g)(x)

(26.17)

k=0

(

~

} X

~dx

' f(x)g(x))

=

(f*g)(x)

n

(26.18)

= = («): (aj^^f-DviyV/jyifix))) x ^(-D e;(y ( x- (y (f(x))) 1

a+P+n+1

k

n

k

a

k

k = 00

x

under

(^_] k (x a+n- 1 f^l n (x- a -^ (x )))

conditions: ReW Reifi)

<< --—\ ,,

Re(a+0) > - — , ReU+P)

2 s g n ( c . ) + s g n ( yy . - nn)) £ * 0,

(26.19) i = 1,2, n e I N ,

a * 0, Re(oc) Re(a) >

238

i

i

-—. 2

5.

If

H(x) = r(l-a-/3-T)/r(l-/3-x),

then

(x^" a x a,3 )(f(x)g(x)) = (f*g)(x) (26.20) 00

P k a a p

- I (:)(■*)<•<(x))(x I - x - )(f(x)). 5

k=0

where

are generalized

ReU-fS)

binomial

coefficients

under

conditions:

> 0, Re{(3) < 0,

(26.21)

+ sgn(

^^-{wzY

V Re{ S }} * °- i = {2}'

6. If H(x) = r(l-a-^+y/2+T)/r(l+y-a-^+x), then

(x1+9r/2-a-|3I3r/2xa+/3^-1)(f(x)g(x)) = (f*g)(x)

(rd + r/2)) 3 00

(26.22)

x )

' ' t r ^ ) ( y / 2 + k)2

7

r/2,l-a

k!

i.y/2

«- T / 2 -x

l+y/2+k,l-y/2-k

k=0

, y/2+1-£ l+y/2 ^-y/2-1 , , > (x F x g) (x) l+3r/2+k,l-3r/2-k °

under

conditions:

Reiy/2-oc-p)

>--?-,

Ke(y-a-0) > - - ? - ,

Re-j^}- > \- ,

(26.23) 2 s g n ( c - — ) + s g n ( * . - 1+ Re{y)/2 i

2

239

+ KejjH) - °» i = - u } -

7. If H(x) = r ( r + 5 - a - 0 - l - T ) / r ( S - 0 - T ) , (x' (26.24)

I

a

xa

then

) ( f ( x ) g ( x ) ) = (f*g)(x)

+oo

V^ fy-a-ll , i-5Tn+a 5 - a - i w _ , . w £ ..1-7-11 y - £ - i w , x. = > , (x I+ x ) ( f ( x ) ) ( x ' I+ x° ' ) ( g ( x ) ) + n=-oo

under

conditions: Re(y+S-a-/3) > 2, Ke(S-/3) > 1, Re(5-a) > — , 2

(26.25) ite(a) > 0,

2sgn(Ci - \ )

a * IN, fle(/3)+ —

* IN,

Re{y-{3) > — , * Re(fS) > - — , fle(S) < — , Re(y)

< 2,

+ sgn(yi+ *efcjl + 1) * 0,

i = jU-

8. If H(x) = l/(r(a+/3+T)r(l-a-0-x)), then (xa+P|(l+x)'1|x"a"P)(f(x)g(x)) = (f*g)(x)

(26.26)

= J(x a {(l + x)-j~x-^ +

under (26.27)

(x 1/2+a {(l + x)-j"x- 1/2 - a )(f(x))(x^(l + x)^

conditions: -—

< Re{a+p) < 0,

2sgn(c -1) + sgn(y ) £ 0, i = 1,2.

9. If H(T) = l/(r(S-a+T)r(y-0-T)), then

(x5'a|r(5+r-a-^)(l+x)a+P"5"y| xa"5)(f(x)g(x)) = (f*g)(x)

240

]

= rty^-a-g-D 2 ^ ( x |n8-g+l)(l+x)

(26.28)

jx

(f(x))

n = -oo

x

under

(x1-a{r(y-«+l)(l+x)a-y"irIxa+n-1)(g(x))

conditions:

(26.29)

fie(a)- — <£ I, 2 Re(8-oc) > - — ,

Re{(3)+— £ Z, ' 2 Re(?-/3) > 1, /te(5-|3) > 0,

fte(a+0-y-5) < - 1 , 2sgn(c.-l) +

Remark (26.20),

26.1.

We

note

that

■*><*,♦ ^ a } ' ^ °- i = f2}-

formulas

(26.24) can be easily reduced

for fractional

(26.15),

shall

demonstrate

the

(26.17),

(26.18),

to known classical Leibniz

integro-differential operators

Proof. We

Re(y-a) > 0,

typical

rules

(see references above).

moments

these relations on the examples of the formulas

in

the proofs

(26.20) and

others are proved in the same manner. At first we will prove

Let us take a = -a, b = t, c = 1-/3-S in the formula

of

(26.24).The (26.20).

(26.2). Then we

obtain the following representation:

r(i+a-/3-s-t) _ rq+q-ft-s) ra-/3-s-t) ni-0-s) 2i

-a, t; l-/3-s;

(26.30) r(l+a-/3-s) r(i-|3-s)

I k=0

Since the gamma-ratio ^ of #

this

theorem

then

(-a) (t) k

k

Re(oc-fS)

> 0.

k! (1-/3-S) k

(s+t) = H ( s + t ) , where H(s) is defined by case 5

the

G-convolution

(t) = 1, takes the next form

241

of

type

(23.3),

where

# (s) =

(26.31)

P

(f*g)(x) = = — —wl+o P ?/) f-t) f*(s)g*(t)x-s f*(s)g*(t)x-s Sisdt. \isdt. (f*g)(x) T[!+o 1 _ P3 3 (27ii)2J J r' (27ii)2J J r ( 1-/ 's "-^ t) (T (T t s

It follows from Theorem 23.1 that G-convolution conditions

(26.21) and the double

integral

(26.31) exists under

is equal

to the repeated

integral. Hence similarly (20.13) we have the representation

T ) ( ) x T ddT, xl/2(f*g)(x) = U J nr }!; gn- T^ ; 7F FTT)*~ xl/2(f*g)(x) = —^2712 " T> rci/2-^-x) (

P

(26.32)

(T (T

where

TT

F(x) = -- U U f*(x-tt— )g*(t)dt

(26.33)

27li

T

for

T T


2 t

Further, from the Theorem 15.3, the Mellin correspondence 9 from the Table

18.2 and

the Definition 20.2

it

is not difficult

to get

the

following equality

(f*g)(x) = (x/3IVxa_/3((f(x)g(x)).

(26.34)

++

Now using

the representation

(26.30) and

substituting

the series in

(26.31) we have:

(f*g)(x) (26.35)

+oo

k =. _- Ji _ [f [ rE f - o g - c u dxsdt. -w "( 1i;T; *y:y^ fY (-a)k k(t)k f*(s)g*(t)x— (27ii)2 J J J J (2Tti)2

r(1

cr
We

will

show

that

"^-s) "P-s)

one

integration and summation

can

LLk = 0

k=0

k!(l-p-s) k-d-p-s) k

change

k

the

order

in the right part of

of

operations

of

(26.35) by means of

Lebesgue theorem. At first, we estimate the partial sum S (s,t) of the series 242 242

s(s,t) = p T ^ y—-—^f*(s)g*(t)x-s-t . 1Z (-a) (t)

n

(26.36)

S(s,t) = r T ^ 7 T 1U li SJ

^f*(s)g*(t Ix" 5 ^

"

I— k!(1-0-s) k=0

k

Re is) = — , Reit) Re{t) = — , x > 0. It is convenient to represent when Reis) S (s,t) in the form: N

>t) SN( - r(i-p-s) N( ss,t) rd-p-s)

s

L ,(i-p-s) 2- kk-(i-p-s) k =0

k

Jt (-a) (t)

(26.37)

+

s ^ ^?Z?^ 7£h^ ± fi * t)x ",x~ " t m f (' s( , 8g ,( g (t " 't

k=N

1

'

k

= S (s,t) + S'(S,t), S'(s,t), N

N

1

where N is a constant, which we will define in the further discussion. At first, we estimate every term s (s,t) of the finite sum S k

(s,t),

N

1 using tne ioiiowing simple consequences o± tne biining s iormuia. and using tne ioiiowing simple consequences o± tne btining s iormuia ana

the Definition 15.2: Re
(( 22 66 -. 33 88 ))

| \ v^\ g Wj C| s \

(26.39)

k k |™*>| |r[^k)[ = |t| |t| [i|\ ♦+o0d(/l |/t| |t )| )] j, ,

(26.40] (26.40)

II ((-_(X a )) I

k!

I k!

I I

k - s -s_t t

x"

|

(26.41)

*

y y nc I s II

* f

(s)|s|

l

e

1

a f ( s ) | s | *2*e"c t g (t)|t| 2e 2

g (t)|t|

e

'

i fi f I t| tI |—> —>0000, , t t €€(T(T. .

i f Re(s) = _L , Reit) i f Reis) = — 2 , Reit)

< C,

I

if IS I ^ —> « 00 ,s , S G ,.. (T . if |s| 6 1

=

F ( s ) 6 L(cr ) , s

= F ( s ) e L(cr ) , = G ( t ) € L(
= — , x > 0. = — 2 , x > 0.

s

Then we obtain, that |s (s,t)| < C |F(s)G(t)| and

(26.42)

IS 1

N

(s,t)| < C |F(s)G(t)| e L(tr x (r ). '

1

3'

'

t

s

For estimation S'(s,t) we will use the following simple inequalities:

(26.43)

|T(x) I * TiReM)

(26.44)

|B(x,y)|

if Reix)

,

r(x)r(y)

> 0.

< B(/te(x),/te(y))

f(x+y)

if Reix)

> 0,

Reiy)

> 0.

Then we have in particular:

(26.45)

|r(l+a-/3-s)r(-a+k] T(l-/3+k-s)

r(l/2+/te(a)-/te(f3))r(-/te(a)+k) r(l/2-/?e(|3)+k) if k > Reoc,

(26.46)

|r(t+k)I * r( — + k ) ,

k = 0, 1,2,

k = 0,1,2,..,

| g (t)/r(t)| ^ c G (t)

(26.47)

where G (t) e L{cr )

If we choose N true for any k £ N

(see the conditions (26.21)).

= max{l, [/te(a)]+l>, then inequality (26.45) holds and using (26.43) —

(26.47) with conditions (26.21),

the next estimations take place

|s;(s,t)| =

rd-/3-s)

( a)

k(Uk

Y—-

r(l+a-0-s) Vx

l—

* . * . . -s-t -f (s)g (t)x /3-s)

k!(l-

* If (s)|| g (t)/r(t) x

f(-a)

244

r( —+/te(a)-Re(|3))

V^ r(-/te(ct)+k) r (1/2+k) l^ r(l/2-Re(p)+k ) k!

(26.48)

I

s Cs|F(s)Gi(t)|

-Rect+Refi-l

> k

C | F ( s ) G ( t ) | e L(cr x
1

'

t

s

Finally, in view of (26.37), (26.42), (26.48),we can apply the Lebesgue theorem and rewrite (26.35) in the form:

(f*g)(x)

r(l+a-|3-s)(

I—

a)

kU)k

*

r(i-p-s) k!d-p-s) f

. * .. - s -t. ,.

(s)g (t)x

dsdt

t— (2wi) cr cr t s

(26.49)

where

the

operators

I

(-a)

£

[ I )(x*"k Ik"a x a - /3 )(f(x))(x k (^) k )(g(x )),

k!

last

k

1

r(l+a-/3-s) _ _ ^ _ _ _ f *( s ),x -s, ds x

2ni

equality

as G-transforms

1

(t)

g (t)x" clt

2TTI

follows in

the

from

the definitions

Table 17.1. Comparing

of

fractional (26.34) and

(26.49) we conclude, that Leibniz rule (26.20) holds true under condi­ tions (26.21).

We will prove the formula (26.24) now.

Let us take a = a, b = /3+t, c = r, d = 5-s in the formula (26.5). Then we obtain the following representation:

245

r(y+5-l-a-/3-s-t) _ sin(7ra)sin(7cQ+t))r(y-a)r(a-a-s)r(y-/3-t) r(5-|3-s-t) (26.50) y /

x

The corresponding

G-convolution

r a+n,/3+t+n 1 |_ y+n, 5-s+n J

(23.3)

^ ( ^ 5 . ^ ) > 2.

(see case 7 of

this

theorem)

takes the form

(26.51)

(f*g)(x) =

r(5-g-s-t)—

(27ii)2

f (s)g (t)x

dsdt

-

cr cr

t s

It follows from Theorem 23.1 that G-convolution

(26.51) exists under

conditions (26.25) and we can represent it in the next form

(26.52)

(f.g)(x) =

(x0-S+V

+a

- V r + a - a - * - 2 )(f(x)g(x)).

Substituting the representation (26.50) in (26.51) we have

(f*g)(x) =

sin(7ia)sin(rc(f3+t) )r(y-a)r(5-a-s)r(y-/3-t)

1 {ZniV t s

(26.53)

\^ _f a+n,£+t+n 1 *, , *,., -s-t, ,, ) T ' f(s)g(t)x dsdt. & / [ y+n,5-s+n J

We

will

show

that

one

integration and summation

can

change

the

order

in the right part of

of

operations

of

(26.53) by means of

Lebesgue theorem. For the sake of brevity we introduce the following notation:

246

,

h

t)

n

sin(Tra)sin(Tr(/3+t))r(y-a)r(5-g-s)r(y-/3-t) 2

I" a+n,/3+t+n 1 [ 7f+n, 5-s+n J

x f^slg^Dx"5"1.

At first, we estimate the partial sum S (s,t): N

n

(26.54)

S N (s,t) =

) hk (s,t),

Re{s)

if

L.

= 2— , fie(t)

, x > 0.

It is convenient to represent S (s,t) in the form: N

N

-1

(26.55)

S (s,t) N

= £\(s,t) + £\ s,t) + ) h

k

k=-N

/ *—' k=0

1

k

s,t

= S

N 1

s,t

+ S'(s,t). N

We estimate the sum S'(s,t), using the following simple relations in N

accordance with conditions (26.25):

(26.56)

r(y-(x)sin(7ra)

-s-t X

2

if Re{s)

l'

Re(t)

— , x > 0

A

2

(26.57)

r(a+k)r(<5-a-s) T(5-s+k)

is a constant. 1

r(ftea+k)r(Re(6-a)-l/2) r(tfeS-l/2+k)

if Re (a) > 0,

Re[8-a)

> — , k

0,1,2,

2

(26.58)

|TO+t+k)| * r(ReO)+-i-+k),

(26.59)

|f (s)| < F 2 (s),

if Re(fB)

where F (s) e L(cr 2

247

s

> - —

(26.60)

|r(r-0-t)sin(ir(0+t))g (t)| * A G (t),

where G ^ t ) e LCo-J.

Then, we obtain that

|S'(s,t)|

V h (s,t) * ) h (s,t)

1

L

N

'

■I

(26.61)

k

\ L\ k

r(y-a)sin(7ra)

r(r+k)

-s-t

r(a+k)r(5-a-s] |r(/3+t+k)| r(5-s+k)

If (s)| |r(y-0-t)sin(nO+t))g (t)|

/2+k) V" r(Jte(<x)+k)r(/te(g) + l * A F (s)G (t) l^ r(Re(5)-l/2+k)r(r (y+k) 3 2 2

But the last series in (26.61) converges under conditions (26.25) since

k*%'vRe(oc+(S-8-y) '" ~ B'+1fl + 0(l/k)

r(/te(a)+k)r(Ke(l3) + l/2+k) = = r(fie(S)-l/2+k)r(y+k)

Thus

S'(s.t) 1

N

< A '

F (s)G (t) 4' 2

2

For estimating the sum

]•

€ L(
t

s

S (s,t) we represent it in the form: N 1

5 (s t) = V 5

'

N

sin(7ry)sin(7r(5-s))r(?-a)r(5-a-s)r(?-/3-t

/

2

(26.62) _f l-y+k,l-S+s+k "I *

. *

r[ x-^.x-p-t+k Jf (s)g Similarly,

we

use

the

corresponding

.-

(t)x

relations

for

gamma-functions

and functions f (s), g (t), which are provided by conditions (26.25). Hence, it is not difficult to establish the next estimation 248

|S ( s , t ) | * A 5 | g ( t ) | | r ( 5 - a - s ) s i n ( w ( 5 - s ) ) f

(s)|

1

V"

(26.63)

r(l-Re{y)+k)r{3/2-R -/te(S)+k) a+k)

l^

r(l/2-/?e(|3)+k)r(l

k=l oo

< A F (s)G ( t ) 6 3

)

n

€ L(cr x cr ) .

3

Thus, u s i n g L e b e s g u e theorem we can r e w r i t e

) (y+n)

1 2711

n=-oo

2711

(26.53)

in the

form:

r(S-a-s) * , -s, =7^ -—c— f ( s ) x d s f(5-s+n)

r(y-/3-t)

* .,

-t,.

ra-p-t-n)

g (t)x

dt

(26.64)

E(^n:i)(xl"5"ni"ax5"a"1)(f

(x))

, (B+n T l-9T-n r-fB-1 w , x x X (x' I X ) (g(x)) .

Comparing

(26.64) and

(26.52) we conclude, that Leibniz rule

(26.24)

holds true under the conditions (26.25).

Similarly,

using

the

definitions

1) — 5 )

of

G-transforms

and

summation theorems (26.2)—(26.5) we can establish the other Leibniz rules in cases 1 — 4 , 6, 8, 9 of the Theorem 26.1, namely:

1) applying addition formula for gamma-function (26.1) we can obtain the rule (26.26);

2) in Gaussian summation theorem

(26.2) setting a = a+s, b = t+/3,

c = y we can obtain the rule (26.9); setting a = a+s, b = /3-t, c = s+y we

249

can obtain the rule (26.13); setting a = -n, b = -t-0, c = s+a, we can obtain the rule

(26.17); setting a = -n, b = t+0, c = 1-n-s-a we can

obtain the rule (26.15); 3) in Saalschiitz's summation theorem (26.3) setting a = s, b = -t-0, c = s+a we can obtain the rule (26.18); 4)

in

c = -t+0

Dixon's

summation

theorem

(26.4)

setting

a = y,b = -s+a,

we can obtain the rule (26.22);

5) in Dougall's formula (26.5) setting a = a, b = f, we

can obtain

the

rule

(26.11);

setting

d = 6+s we can obtain the rule (26.28).

Remark

c = y+s, d = 5+t b = 0+s,

a = a-t,

c = y-t,

■

26.2. The others series representations

of

G-convolution

(23.3) can be received, for example, from the values of hypergeometric functions in particular points (see A.P.Prudnikov et al. (1989)). The formulas (26.6) —

(26.8) can be applied for obtaining the integral

analogs of Leibniz rules for some G-transforms.

Theorem 26.2. Suppoee = HH(x). ( T ) . Then 1 1. If

the

following

f e Wf 3H"11

that integral

analogs

(L), g € 3 3JT J1 *1 of

Leibniz

(L) and

rul2s

H(Th = the(a+0-l+x) the ( CL+B— 1 + T ) tien tien H(Th

( xa+/aR+ / V x (x V x (26.65)

R '

'

)(f(x)g(x)) = (f*g)(x) )(f(x)g(x)) = (f*g)(x)

+ 00

_ 22-a_/3

(xa+TA _1x"^"Tf) ( 2x) (xa" T A _1 x~^+T

)(2 Id

-00

under (26.66)

conddiions: Re(a+0) > 1,

2sgn(c -1) + sgn(y ) £ * 0, i = 1,2. i

250

i

hold

* - 1 (x) true:

2 . If

H(x) = r ( a + | 3 + y + 5 + l - T ) / r ( | 3 + 5 + l - T )

then

, -/3-5 T -a-3r a+/3+3r+5 w _ r . , . . _ . (x I x ) ( f ( x ) g ( x ) ) = r( f * gw) ( x ) (26.67) Ia+3^| * -^-x T T-3r £ + 3 ^ , ^ , -<5+TT-T-a a+5 w . , I (x I °x' ° f ) ( x ) ( x I x g)(x)dx

under

conditions: Re{oc+(3+y+8) > 0,

(26.68)

fie(a+S)

Re{(3+8) > 0,

> - —,

ReO+y) > - — ,

iiB+ ]

2 s g n ( c - — ) + s g n ( * - Re\ 1 2

3 . If

i

H ( T ) = r(a+/3+r+S-l+x)

, a+^+y+6-i.

(x (26.69)

Ax

*

0,

i

(a+s: then

-a-^-y-5+i

_. . , . .

w

,_

.

) ( f ( x ) g ( x ) ) = ( f * gw) ( x )

+ oo

—,

1

ww^

, oc+tTy-8+i - a - 3 r _ w

r(x

r(^-x)r(5-T)

under

Re(a+y) > - 1 ,


.

X

T-6+1

w

'f)(x)(x'

fi+8r8-y+i *

x-^+i

-(B-8

X

conditions: linx. min^Re(a+y) , R e ( a + 5 ) ,Re(/3+*) , Re(/3+S) V > - -

( 2 6 . 7 0 ) Re(a+fi+y+8)

> — , 2 s g n ( c )+ s g n ( * - Re 2

i

i

2f

251

(2a+*+S) (2/3+r+6)

) *

w

._.

g)(x)dT

°

0,

4. If

then

H(x) = l/(r(a+5+l-x)rO+y-l+x))

(x

(26.71)

^ + 3r-i| r(a+ ^ + ^ +g)(1+x) -a-^-2r-6| x - / 3 - y + l ) ( f ( x ) g ( x ) )

=

(fWg) (x)

13 5 1 _ r(/3+5)r(a+y) (x/3+T|r(/3+5+l)(l+x)-|3-5-1U^-T)(f(x)) r(a+/3+y+S-l)

^jro+s+Dd+x)- - - }"^-

(x*"T{r

x (x<,"^r(a+y+l)(l+x)"a"9r"1i xT_3r)(g(x))dT

under

conditions: Re(a+/3+?+S) > 1,

( 2 6 . 72)

/te(|3+y) >

min^ nirw.Re ( a + 5 ) , R e ( a + * ) , Re(|3+5) 2sgn(c.-l)

+ s g n ( * . ) £ 0,

5. i f H ( T ) = r(a+£+x)/r(a+/3+ — +T)

—,

}>o,

i = 1,2.

then

, a+/3Ti/2 -a-0-i/2Wx., > , x, (x I x )(f(x)g(x), (26.73) , a-ixA1 (x Al 271

under

x

-a+ix _ w w 0-iXA. f)(x)(x' Al

2iX

-0+iX w » , x' g)(x)dx 2iX

conditions: Re {a) > - — ,

Re(a+/3) > - — , 2

(26.74)

2sgn(c -1) + sgn(? -1) ^ 0, i

Re({3) >

2

i = 1,2.

i

Proof. For example, we consider

the proof of rule

(26.65).

The

others are proved in the same manner. Let us take c = s+a, d = t+/3 in the formula (26.7). Then we obtain the following integral representation: 252

(26.75)

1 = 22-oc-B-s-t T(a+£-l+s+t]

dx r(s+a+x)r(t+/3-x) '

Reiot+p)

> 0.

The corresponding G-convolution takes the form

(26.76)

1

(f*g)(x)

r(«+g-l+5+t) f Cs)g (t)x-s"dsdt.

(2iri)'

It follows from Theorem 23.1 that G-convolution

(26.76) exists under

conditions (26.66) and by equality (18.3) we can represent it as follows

(26.77)

(f«g)(x) = (xa+/3 1A lx

a /3+1

)(f(x)g(x)).

Now we use the integral (26.7) for the kernel of G-convolution

(26.76)

and obtain: * . * . -s-t 02-a-/3-s-t 2 f (s)g (t)x

(f*g)(x) (2712)'

(26.78) &z dsdt. r(s+a+x)r(t+/3-x]

We

will

show

that

one

can

change

the

order

of

operations

of

integration in the right part of (26.78) by means of Fubini theorem. For the sake of brevity we introduce the following notation:

(26.79)

F(s,t,x,x) =

2

f (s) g (t) x r(s+a+x)r(t+/3-x)

We consider the following integral now

(26.80)

I(x) =

F(s,t,x,x)dxdsdt. (T (T -00

t s

253

It i s convenient to represent

I(x)

i n t h e form

F(s,t,x,x)dTdsdt +

I(x) =
F(s,t,x,x)dxdsdt cr (T - N t s

(26.81)

F(s,t,x,x)dxdsdt

= I (x) + I (x) + I ( x ) , 1

2

3

CT (T N

t s

where N is a constant, which we will define in the further discussion. For estimating the integral I (x) we have |l 2 (x)|


|F(s,t,x,x)Idxdsdt +

|F(s,t,x,x)Idxdsdt

|F(s,t,x,x)Idxdsdt +

|F(s,t,x,x)Idxdsdt

v
(26.82)


v >r -N

I

(x) + I

21

>r (x) + 1

22

where s = — + iu,

(x) + I (x), 23

t = —+iv,

2

v >r -N

24

r is a fixed constant.

2

Hence we must show that I2j (x) < +oo, j = 1,2,3,4. This fact follows from the conditions

(26.66) and the next estimations of integrand in

each integral (26.82):

for I (x): 21 2-a-/3-s-t

0 2

'

*

.

*

r(s+a+x)r(t+/3-x) where B

.

-s-t

f (s) g (t) x

< B If*(s)g* (t)| 6 L(cr xcr x(-N,N)), 1'

is a constant,lul < r,IvI < r;

254

'

s

t

for

I

(x):

2-a-/3-s-t * . * . -s-t 20 f (s) g ( t ) x r(s+a+T)r(t+£-T) where B

for

is a constant,

I

|u|

> r,

^ _ If


(s) g ( t ) I

Hs+a+r)

v

< r;


3i

. ,

,

M KI.

.

1e L(VV(-N,N)).

(x): 23

2

' f (s) g ( t ) x T(s+a+T)r(t+^-T)

where B

f

rit!AS t ) l

*UVV(-N.N)).

is a constant, |u| < r, |v| > r ;

for I (x) 24

* . * . -s-t f (s)g (t) x r(s+a+x)r(t+/3-T)

02-a-/3-s-t

2

where B

< B

f * (s)g* (t) r(s+a+x)r(t+/3-x)

€ L(or xcr x(-N,N)), s

t

is a constant, |u| > r, |v| > r.

Then we will estimate I (x). l

(26.83)

|F(s,t,T,x)IdTdsdt

I (x) l

|F(s,t,-T,x)Idxdsdt. cr cr N

(T (T -co t s

t s

Hence, using formula (26.1) we obtain the following representation:

|F(s,t,-x,x)

22"a"/3r(l-s-(x-HT)sin(7i(l-s-a+x))f*(s)g*(t)(2x)"s"t T(t+/3+x)

Thus, using inequality (26.44) we get 255

22"a"/3r(l-s-a+T)sin(7r(l-s-a+T))f*(s)g*(t)(2x)"s"t r(t+0+x)

*

^ T(l/2-Re(a)+T)- . , ,„ , * , , ■ g (t) i r(l/2^e(3) + T)l S i n ( 7 r ( 1 - S -" + T ) ) f (S) I r(t+/3+A)

C

where A is a constant and A > - - - Re(6), 2

- -fte(a)+x > 0. Hence, 2

|F(s,t,-T,x)|
g (t) f(t+/3+A)

€ L(cr x
t

if we choose N = max{l, Re(oc)-—

, - — -Re(ft)}. 2

can be obtain

in the same

The estimation of I (x)

2

manner

3

as I (x), if N = max{l, - R e ( a ) - — , 1

-_L

+

2

Re(/3)>.

Thus

if we

- — +/te(/3)>

choose

Re{oc)-

N = max{l,

— -/te(0),

-Re(a)- — ,

2

then all integrals in the right parts of (26.82),

2

(26.83)

converge and using Fubini theorem we can rewrite (26.78) in the form

(26.84)

^2-a-0

(f*g)(x)

(x

Comparing

similarly

A

x

other by

integral using

g (t)(2x)'

ru+p-x)

A

x '

that Leibniz

dtdx

g)(2x).

rule

(26.65)

(26.66).

analogs

the

1 27ri

f)(2x)(x'

(26.77) and (26.84) we conclude,

holds true under conditions

The

f (s)(2x)" s ds r(s+a+r)

1 2?ii

in this

formulas

definitions of G-transforms, namely: 256

theorem

can be

(26.6)—(26.8)

and

established corresponding

1) in integral (26.6) setting a = a+1, b = /3-s+l, c = y+1, d = 5-t + l we can obtain the analog (26.67); setting a = a+l+s, b = /3+1+t, c = y, d = 5 we can obtain

the analog

(26.69);

setting a = a+l-t, b = £+s,

c = y+t, d = 6+1-s we can obtain the analog (26.71);

2) in integral

(26.8) setting a = s+a, b = t+/3 we can obtain the

analog (26.73). Finally

we will

formulate

algorithm

for

the

evaluation

of

some

classes of integrals and series with respect to the indices (parameters) of

hypergeometric

functions

as

the

application

of

the

Theorems

26.1 — 2 6 . 2 .

ALGORITHM FOR THE EVALUATION OF INDEX INTEGRALS AND SERIES

First we derive some corollaries from the Theorems 2 6 . 1 — 2 6 . 2 . If functions f(x) and g(x) in these theorems are Meijer G-functions (1.3) then we have the following statements:

1) one can verify the conditions in these theorems without any difficul­ ties;

2) the result of applying G-transform (17.1) to Meijer G-function is the new G-function;

3) G-convolutions

(23.3) of Meijer G-functions are G-functions of two

variables (13.1).

For example, Leibniz type rule (26.9) will have the following form

257

I

1+/3-?, (c ) , 0

1+a—y,(a ) , a , r + l , l + l J p u+2,v+l a+k,(b )

1 _m+l,n+l r ( y + k ) k !.G p+2,q+2

u

0+k, (d )

q

(26.85)

l+a+0-y:(a );

0, l : m , n ; r , 1

= G

p

1, 0 : p , q ; u , v crx, wx

: (b ) ; q

(c )) u

(d ) v

under these conditions: Reiv-oi-fi)

> 1,

2sgn(c.) + sgn(y.-l)

Re {a)

Re((3)

> -■

> 0,

>--!-,

i = 1,2,

{;}»-»• >-©•

2sgn(c.) + sgn(y.-Re(y-2-

where pairs (c ,^ ) are defined by (17.4) i

i

The same corollaries have been obtained

for others Leibniz type

rules and their integral analogs.

Then the algorithm for the evaluation of index integrals and series is divided into four steps:

1. Representation of functions in the integral or sum in terms of Meijer G-functions.

2. Search for the appropriate Leibniz type rule of the form such as (26.85).

3. Representation of index integral or sum in terms of G-function of two variables.

4. Representation of G-function of two variables by means of more usual special functions.

We illustrate this algorithm by the following example: one needs to evaluate the given sum: 258

k!

q = V

p(y-i,a-a-k)f

^r(^+k) k where P

I

)

2

!+ bx J

(y-i,c-|3-k)( k

I

_ 2 _ | + dx

1

J'

(z) is the Jakobi polynomials (A.Erdelyi et al. (1953)).

1. We use the following representation in terms of G-function

(y-i,a-a-k)( k [

2 1 + bxJ

_ (Ubx) a t Y ' a (bx)" a 2,l bx r(a+y-cc) k! 2,2

l+a-y,a a+k, a

under conditions: Reia)

Re(y-a) > — ,

> -— ,

/te(c) > --

2

/te(a) > - — ,

/te(0) > - —

2

2. One can see that this sum may be evaluated by means of the Leibniz rule (26.85) under condition

Re(y-a-/3) > 1.

3. We

the

obtain

the value

of

sum

in

terms

of

G-function

of

two

variables:

S =

(l + bx) a + *" a ( l + d x ) c + y - * T(a+y-a) r(c+y-/3)(bx)a(dx)°

l+CL+(S-y:

^0,1:1,0;1,0 1,0:0,1;0,1

bx.dx : a ; c

4. We represent

the resulting G-function

of

two variables using the

property (13.7) in terms of more simple function:

S =

(l+bx) a+ *

a

(l+dx) C+3r ^ r(y+a+c-arCa+y-a) r(c+y-/3) (1+bx+dx)3r+a+c-a-£'

259

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Author index

Agarwal, R.P. 5,12,102,261 Aizenberg, L.A. 57,261 Anguino M.E.F. de. 261 Appell, P. 3,116,261

Bailey, W.N. 5 Barnes, E.W. 1 Bora, S.L. 6,12,74,261 Bozhinov, N. 261 Braaksma, B.L.J. 3,261 Brychkov, Yu.A. 2,145,261,272 Burchnall, J.L. 4,5,262 Buschman, R.G. 8,9,10,11,12,13,22,45,132,262,263,271,274,277

Chaturvedi, K.K. 6,12,74,263 Chaundy, T.W. 4,5,262 Cih, A.K. 57,263

Daoust, M.C. 103,274 Davis, B. 263 Dimovski, I.H. 162,190,264 Ditkin, V.A. 163,264,265

279

Dixon, A.L. 12,132,265 Doetsch, G. 265 Dzrbasjan, M.M. 265

Embrechts, P. 265 Erdelyi, A. 1,4,24,41,43,65,84,93,112,113,116,163,217,218,259,265 Exton, H. 5,265

Fedorjuk, M.V. 265 Ferrar, W.L. 12,132,265 Flensted-Jensen, M. 162,265 Fox,C. 2,144,265

Gauss, C.F. 1 Glaeske, H.-J. 145,162,261,266 Goldberg, R.R. 266 Goyal, A.N. 6,12,74,263,266 Goyal, S.P. 3,8,11,266,274 Gupta, K.C. 3,7,8,11,12,71,132,263,266,267,270,274

HeB, A. 162,266 Hirchman, I.I. 162,267 Horn, J. 3,59,267

Joshi, C M . 267

280

Kakichev, V.A. 162,267 Kalla, S.L.

6,12,74,261,264,267,270,271

Kampe de Feriet, J. 4,103,113,261 Kapoor, V.K. 144,267 Karlsson, P.W. 5,86,274 Kesarwani, R.N. 144,268 Kilbas, A.A. 220,233,273 Kiryakova,

V.S. 190,264,268

Koornwinder, T. 162,265 Koul, C.L. 263,274 Kratzel, E. 268 Kummer, E.E. 1

Luchko, Yu.F. 233,268,277 Luke, Y.L. 1,2,92,268

Magnus, W. 265 Marichev, 0.I. 2,11,15,43,58,71,82,85,92,104,116,119,120,141,144, 217,219,222,230,233,261,268,269,271,272,273,276 Masood, S. 144,267 Mathai, A.M. 2,3,270 Meljer, C.S. 2,270 Mellin, Hj. 1 Mittal, P.K.

7,12,71,132,266,267,270

Mourya, D.P. 270 Munot, P.C. 6,12,74,270

281

Nguyen Thanh Hai

4.104.116.184,270,271,272,277

Nieva Pino, M.E. 271 Nikiforov, A.F. 271

Oberhettinger, F. 265 Omey, E. 265 Osier, T.J.

233,271,272

Pathak, R.S. 6,12,74,272 Pincherle, S. 1 Prajapat. M.L. 267 Prasad, Y.N. 272 Prudnikov, A.P.

Raina, R.K.

2.84,92,93.145,156,163,223,233,250,261,264,265,272

274

Rathie, N. 272 Reed, I.S. 55,272

Saigo, M. 116,272,273 Samko, S.G. 220,233,273 Saxena, R.K.

2,3,74,270,273

Saxena, V.P. 4,273 Shah, M. 6,12,74,273 Sharma, B.L. Singh,

5,12,102,273

A.K. 272

Singh, R. 132,273 Slater, L.J.

1,43,58,68,86,233,273

282

Sneddon, I.N. 162,225,273 Srinivasan, V. 274 Srivastava, H.M. 3,5,8,11,22,86,103,104,132,263,271,274

Ta Li 274 Tandon, O.P. 22,275 Tanno, Y. 275 Titchmarsh, E.G. 119,121,131,162,275 Tricomi, F.G. 265

Uvarov,

V.B.

271

Verma, R.U. 6,12,74,275 Vilenkin, N.Ya. 162,275 Vu Kim Tuan 11,13,71,120,132,145,222,261,269,275,276

Watanabe, Y. 233,276 Widder, D.W. 161,276 Wiener, N. 162,276

Yakubovich, S.B. 120,144,162,184,222,233,268,271,273,276,277 Yanushauskas, A.I. 109,116,277 Yuzakov, A.P. 57,261

Zemanian, A.H. 277

283

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Subject index

addition formula for gamma-function 233 additional probability's integral 93 Agarwal's G-function 5 algorithm for the evaluation of index integrals and series 257 Appell functions (series) 42,113,114,115

beta-function 41,168,169 beta-integral 193 Bessel function 93,98,99 binomial coefficients 238 Boltz's formula 86 Buschman's H-function of variables 8,9

characteristic of the H-function of two variables 69 Clausenian function 117 composition structure of the G-transform 153 composition structure of the H-transform 140,153 convolution for modified Meijer transform 213 convolution for modified Hankel transform 221 convolution for modified Stieltjes transform 215 convolution for Riemann-Liouville operators 209,210 cos-transform 149 criterion for convergence region of H-integral 13,30,39,40 contiguous relations 47,48

285

degenerate Tricomi function 93,100 Dixon's summation theorem 234 double hypergeometric type series 57 double Mellin-Barnes type integrals 9,170 double Mel1 in transform 54 double Mel1 in transform of the H-function 56 Dougall's formula 234

factorization property 163 factorizations of simplest operators 157 Fox's H-function 3,9,12 Fourier integrals 131 Fubini theorem 121

G-convolution 198,199 G-function of two variables 5,89 G-transform 142,144 G-transform with Gauss hypergeometric function 155,236 Gauss-Legendre product formula 230 Gauss's hypergeometric series

1

Gaussian summation theorem 68,233 general beta-function 193 general hypergeometric function 1 general integral convolution 170 general Laplace convolution 190-193 generalized binomial coefficients 239

286

H-function of two variables 6,7,8,9 H-function of N variables 11 H-function with the third characteristic 72 H-integral 12 H-transform 132 Hankel transform 148,221 Hartogs theorem 51 Horn's list functions 43-46

identical transform 137 incomplete double series 64 index of the G-transform 143 index of the H-transform 133 integral analogs of Leibniz rules 250-252 integral convolutions 161 integral exponential function 93,95 integral transforms with Kummer and Whittaker functions 152 inverse G-transform 143 inverse Mel1 in transform 120 inverse H-transform 136 inverse operator to the operator with Kummer function 236 Jakobi polynomials 259

Kampe de Feriet function (series) 4,103,203 Kummer function 152

287

Laplace convolution 163 Lebesgue theorem 242 Leibniz rules 233,236-241,250-252

Macdonald function 93,99 max-convolution 186,187 Meijer G-function 2 Meijer transform 151,152,213 Mellin-Barnes type of contour integral 2,3 Mellin correspondences 159-161 Mellin-Parseval formula 119 Mellin transform 119,120 Mellin transform of the H-function 134 Mellin type convolution transform 119 Mittal's and Gupta's H-function 7 modified H-convolution 225 modified Laplace transform 146 UJf^L)-space 120 3JT1 (L)-space 121

non-complete gamma-function 93,98 operator with Tricomi function and its inverse 235 operator with algebraic function in the kernel 235 operator with generalized Laguerre polynomial 235

Pochhammer symbol 1,103

288

reduction formulas 80,83-89 Riemann-Lebesgue lemma 121 Riemann-Liouville integro-differential operators 147

Saalschutz's summation theorem 234 series representations 57,217,218 Sharma's S-function 5 sin-transform 149 Slater's notation 43,106 Slater's theorem 58,63,141 Sonine integral 224 Stieltjes transform 151,215 Stirling's formula 110,244 sum and maximum properties 82,93 sum-convolution 183,185

Tuan's generalization of the H-transform 132

Whittaker function 152

289

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NOTATIONS

(a) 1,103 8(s,t) 193 C k 238 n

[ na I 239

I J

(c\/) 143 f*(s) 119,120,121 (f*g)(x) 162,170 L

(f*g)(x) 163 (f*g) (x) 183,185 +

(f*g)

(x) 186

max

(a,/3) (f*g)(x) 211 K V

(f*g)(x) 214 S a (f*g)(x) 216

(a,0) (f*g)(x) 221 sin

(f*g)(x) 225 25 ( f ^ ) ( A , . ) ( x ) 2225

F [x,y; (a,a,A) ; (/3,b,B) ] 59 m n F (ct,p.jS'irj-x,-y) 44,113 291

F (a,j3,0' ;y,y'

;-x,-y) 44,115

F (a,a';£,£';y;-x,-y) 44,114 F (a,/3;y,y' ;-x,-y)

44,114

4

F (a,b;c;z) 152 F (a,b;c;z) 1 2 1

a , . . . ,a i P;

F p q L

b , . . . ,b i

q;

A:B;B' f

>

(a):(b);(b');

4,103

x,y C:D;D' [ ( c ) : ( d ) ; ( d ' ) ;

(a ))

m, n

p.q

z

P q

2 y

:::-[H

p.q

(x) 142

m (3) r(a( 1 ^): r(a( 2 ^ ); ( (a ^ )A |

m ,n :m ,n ;m ,n 1 1 2 2 3 3 p , a : p , q : p ,q x,y *1 n l *2 2 3 3

91

(P ( 1 > ): ( P < 2 ) ) ; ((3<3)) q

q

1

G (a,0,0* ;x,y) m ,n p.q I v

q

2

65

(a ,a ) p

P

' Oq ,b )J qy

292

3

.((a,a)

tfm'n p>q

1,p

O.b) l,q

o["f(u)j(x) 132

y

H[x,y; (a,a,A) ; (0,b,B) ; L J m

n

,LJ

s

t

H[x;(a,a) ;(0,b) ;L ] 9 m n s f

m , n : in , n ; m , n l 1 2 2 3 3 x,y p ,q :p ,q ;p ,q *1

1

2 ^2 *3 ^3

(1), f

(1)

(a

,a

(2)

): (a

(2), .

,a

H (a;r;-x,-y)46H (a;yt5;-x,-y)46 l"(x~af(x)) 147

f (2v£)U|f (u)| 148

}•

f(u)

151

2K (2V^)|o|f(u)l 151

rl" * l ^ ^ a i b j - x j U r f (u)l 152

r\l/2-p-v,l/2-p+v~\e{2x)

xak(x)|o["f(u)l

,a

(/3(1),b(1)):0(2),b(2));(|3(3),b(3))

H (a,0;y;-x,-y) 44,65

r(p)(i+xrpU

(3) ( 3 ) ^

); (a

W

^)}°[ f (u)

145

293

152

) 72

- 1

a

jx k(x)| offCu)! 145

(xVx" a )(f(x)) 146 (xaA;ax"a)(f(x)) 146

( x a * V a f ) ( x ) 235 (xaAl x"af)(x) 235

v

(x a L A x" a f)(x) 235 k

, afr(b)

(X

r

b|

\fUT F a)

X

~ a x-w^^ or>n f H X )

236

(x°VC .x~af)(x) 236 a, b

m jf(x,y); s,t| = f*(s,t) 54

an" ( D 120

m'1

( D 121

L(
t

L(x"1/2;0,co) 185 L , Ln 52 C L L , L. 9,10 s t (Lf)(x) 163 (Sf)(x) 220

j—sin(2\6<)|o['f(u)j 149

294

i—cos(2Vx)lo|f (u)j

149

a ,...■, a i

P

1

q

43,106

#4(a,/3;r,S;-x,-y) * (a»|3;y;-x,-y)

44

45

*2(0,0' ;r;-x,-y)

45

* C p ; y ; - x , - y ) 45 ^(a.jSjy,*' ;-x,-y) *A0C>7>V';-x,-y)

45

46

E ( a , a * , £ ; y ; - x , - y ) 46 J a , / 3 ; r ; - x , - y ) 46 (K,JLI)

133,190

6(T)

8,9

*(x)

3

*(x) 2
295