Shintani-Barnes Zeta and Gamma Functions

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Author: Friedman E | Ruijsenaars S

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ARTICLE IN PRESS

Advances in Mathematics 187 (2004) 362–395

http://www.elsevier.com/locate/aim

Shintani–Barnes zeta and gamma functions Eduardo Friedmana,1 and Simon Ruijsenaarsb b

a Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile Centre for Mathematics and Computer Science, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Received 4 April 2003; accepted 11 July 2003 Communicated by Takahiro Kawai

Abstract We show that Shintani’s work on multiple zeta and gamma functions can be simpliﬁed and extended by exploiting difference equations. We re-prove many of Shintani’s formulas and prove several new ones. Among the latter is a generalization to the Shintani–Barnes gamma pﬃﬃﬃﬃﬃﬃ R1 functions of Raabe’s 1843 formula 0 log GðxÞ dx ¼ log 2p; and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as ‘‘vanishing period integral’’ side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems. r 2003 Elsevier Inc. All rights reserved. MSC: 11S80; 11M35; 11M41; 11R42; 33E30 Keywords: Multiple gamma function; Shintani zeta function; Elliptic gamma function; Raabe’s formula

1. Introduction Motivated by some problems in number theory, Shintani [13–18] introduced in the mid-1970s a multi-dimensional zeta function zðs; M; xÞ; with M an N n matrix M :¼ faij g; i ¼ 1; y; N; j ¼ 1; y; n;

ð1:1Þ

E-mail addresses: [email protected] (E. Friedman), [email protected] (S. Ruijsenaars). Research partially supported by FONDECYT Grant 101-0324 and by the Chilean Presidential Chair in Number Theory. 1

0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.07.020

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with positive entries, xAð0; NÞN ; and zðs; M; xÞ :¼

N X

n Y

N X

m1 ;y;mN ¼0 j¼1

i¼1

! s ðxi þ mi Þaij

;

ReðsÞ4N=n:

ð1:2Þ

For n ¼ 1 this zeta function amounts to Barnes’ multiple zeta function [3]. Shintani showed that his zeta function admits a meromorphic s-continuation with the same pole locations as the function pN;n ðsÞ :¼ Gðns NÞ=GðsÞ;

ð1:3Þ

and obtained various explicit formulas. In particular, he expressed the s-value zð0; M; xÞ and the s-derivative @s zðs; M; xÞjs¼0 in terms of Bernoulli polynomials and Barnes’ multiple zeta and gamma functions. A certain multi-dimensional contour integral yielding the s-continuation played a pivotal role in Shintani’s reasoning. Unfortunately, his impressive calculations rarely gave any insights as to why his formulas should hold or how he had discovered them. A principal purpose of this paper is to present a simpler approach to Shintani’s work. Along the way we also obtain with little extra effort various new results, including the formula Z zðs; M; xÞ dx ¼ 0; I N :¼ ð0; 1ÞN ; ReðsÞoN=n: ð1:4Þ IN

To our knowledge, except when N ¼ n ¼ 1 [4,7], this result is new in the Barnes case, too. A crucial point in our approach is that we mostly work with a zeta function that is somewhat more general than Shintani’s zðs; M; xÞ; namely, ! s N n N Y X X zN;n ðs; wja1 ; y; aN Þ :¼ wj þ mi aij ; ReðsÞ4N=n: ð1:5Þ m1 ;y;mN ¼0 j¼1

i¼1

Here the ai and w are elements of Cn whose coordinates aij and wj have positive real parts. (In Section 6 we slightly relax this restriction.) Thus we have zðs; M; xÞ ¼ zN;n ðs; W ðxÞja1 ; y; aN Þ;

W ðxÞ ¼ W ðxja1 ; y; aN Þ :¼

N X

xi ai : ð1:6Þ

i¼1

Clearly, for NXn the two zeta functions are substantially equivalent, as the ai generically span Cn : For Non; however, the function zN;n ðs; wÞ is more general than zðs; M; xÞ: The key advantage of working with the functions zN;n ðs; wÞ is that they satisfy zN;n ðs; w þ aN ja1 ; y; aN Þ zN;n ðs; wja1 ; y; aN Þ ¼ zN 1;n ðs; wja1 ; y; aN 1 Þ ð1:7Þ

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Q (with z0;n ðs; wÞ :¼ nj¼1 w s j ). This recurrence relation is nearly immediate from the Dirichlet series deﬁnition, and has no analog for zðs; M; xÞ: Viewing (1.7) as a ladder of analytic difference equations, we are able to guess and prove various results, including many of Shintani’s formulas. (In the Barnes case n ¼ 1; the difference equation perspective was exploited before in [11], cf. also [12].) A related point we emphasize is to work directly with the Dirichlet series (1.5) inside its domain of convergence, even though our interest lies mostly in the points s ¼ ðN mÞ=n with m a non-negative integer, especially s ¼ 0: The crux is that formulas obtained via the Dirichlet series for ReðsÞ large can be analytically continued. Shintani worked in the reverse order, ﬁrst analytically continuing zðs; M; xÞ and then manipulating expressions valid at s ¼ 0; but these are more complicated than the series. The simplicity of our ideas might easily remain hidden under the extensive bookkeeping needed to handle the general case. Therefore, we illustrate them in this introduction via the simplest non-trivial case, which is the well-known Hurwitz zeta function (cf. [1, Sections 1.2–1.3]) Hðs; wÞ :¼

N X n¼0

1 ¼ z1;1 ðs; wj1Þ; ðn þ wÞs

ReðsÞ41; ReðwÞ40:

ð1:8Þ

This also serves to explain the organization and main results of Sections 2–5. Our starting point is Euler’s formula Z N e t ts 1 dt: ð1:9Þ GðsÞ ¼ 0

It entails that we may rewrite GðsÞHðs; wÞ for ReðsÞ41 as N Z X m¼0

N

e

rðwþmÞ s 1

r

0

dr ¼

Z

N

0

e rw fðrÞrs 1 dr; r

fðrÞ :¼

N X r al r l ; ¼ 1 e r l! l¼0

with the fðrÞ power series converging for jrjo2p: Obviously, we have fðrÞ

M 1 X l¼0

al r l ¼ OðrM Þ; r-0; l!

fðrÞ ¼ OðrÞ; r-N:

ð1:10Þ

Using (1.9) again, we now obtain GðsÞHðs; wÞ ¼

M 1 X l¼0

al Gð 1 þ l þ sÞ þ w 1þlþs l!

Z

N 0

! M 1 X al r l e rw fðrÞ rs 1 dr: r l! l¼0

ð1:11Þ

Due to bounds (1.10), the integral yields a function that is analytic for ReðsÞ4 M þ 1: From this we easily deduce that Hðs; wÞ extends to a meromorphic function of s; its only singularity being a simple pole at s ¼ 1:

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Of course, this is just one of several ways to obtain these well-known facts. But the present approach admits a straightforward extension to the Barnes case (cf. [11]), and in Section 2 we generalize it to zN;n : The principal result of Section 2 is that zN;n ðs; wÞ=pN;n ðsÞ extends to a function that is entire in s and analytic in the wj and aij in the domains Dn :¼ fwACn jReðwj Þ40; 1pjpng;

ð1:12Þ

DN;n :¼ faij ACjReðaij Þ40; 1pipN; 1pjpng:

ð1:13Þ

and

The associated Shintani–Barnes gamma function GN;n is deﬁned by GN;n ðwja1 ; y; aN Þ :¼ expð@s zN;n ðs; wja1 ; y; aN Þjs¼0 Þ;

ð1:14Þ

@ where @s :¼ @s : Its analyticity features can be elucidated via the ladder of difference equations

GN;n ðw þ aN ja1 ; y; aN Þ 1 ¼ ; GN;n ðwj a1 ; y; aN Þ GN 1;n ðwja1 ; y; aN 1 Þ

G0;n ðwÞ ¼

n Y

w 1 j :

ð1:15Þ

j¼1

In particular, 1=GN;n ðwÞ extends to an entire function on Cn : We now return to our account of further properties of the Hurwitz zeta function, obtained along the lines of Sections 3–5. Accordingly, we will only make use of z1;1 ðs; wÞ=p1;1 ðsÞ being entire in s; but not of formula (1.11) yielding the scontinuation. The entireness property amounts to ðs 1ÞHðs; wÞ being entire, and this is all we need to know to obtain the desired results via the Dirichlet series (1.8) and the pertinent difference equation, namely, Hðs; w þ 1Þ Hðs; wÞ ¼ w s :

ð1:16Þ

As a ﬁrst step, we take the w-derivative of the Dirichlet series, yielding a function @w Hðs; wÞ ¼ s

N X

1

n¼0

ðn þ wÞsþ1

that is clearly analytic for ReðsÞ40: From this we see that the pole of Hðs; wÞ at s ¼ 1 has a constant residue. Next, we observe that the formula @w2 Hðs; wÞ ¼ sðs þ 1Þ

N X

1

n¼0

ðn þ wÞsþ2

yields a representation of @w2 Hðs; wÞ valid for ReðsÞ4 1; whence it is clear that @w2 H vanishes for s ¼ 0: Therefore, Hð0; wÞ is a polynomial of degree at most 1. Taking further derivatives, it follows more generally that Hð k; wÞ with kAN

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is a polynomial of degree at most k þ 1: Since the difference equation (1.16) specializes to Hð k; w þ 1Þ Hð k; wÞ ¼ wk ;

kAN;

ð1:17Þ

we deduce that the degree of Hð k; wÞ is in fact equal to k þ 1: (Here and below, we denote the set of non-negative integers by N and the set of positive integers by Nþ :) In Section 3 we extend this reasoning to the general case, obtaining the polynomial property for several quantities of interest, together with an upper bound on the degree. By additional arguments we then show that the upper bound is optimal. Speciﬁcally, we prove that zN;n ð k; wÞ with kAN is a polynomial of degree N þ kn; and that the eventual simple poles at the s-locations ðN lÞ=n with lAN do occur if laN þ kn ðkANÞ; the residues being polynomials of degree l: Finally, we demonstrate that the difference log GN;n ðwja1 ; y; aN Þ

n X

log GN;1 ðwj ja1j ; y; aNj Þ;

NX1; nX2;

ð1:18Þ

j¼1

equals a degree-N polynomial, and determine this polynomial explicitly for N ¼ 1: Except for the treatment of residues, these results are all due to Shintani [13, Proposition 1 and its Corollary] [16, Proposition 1], albeit for zðs; M; xÞ: The main result of Section 4 is the integral formula (1.4) and its corollary Z log GN;n ðW ðxÞÞ dx ¼ 0: ð1:19Þ IN

For the Hurwitz case (1.4) specializes to Z

1

Hðs; wÞ dw ¼ 0;

ð1:20Þ

ReðsÞo1:

0

We continue by demonstrating (1.20) as a template for the proof of (1.4). We begin by noting that Hðs; w þ 1Þ is analytic for sa1 and ReðwÞ4 1; so that it is integrable in w over (0,1). Speciﬁcally, we obtain from the Dirichlet series (1.8) Z 0

1

Hðs; w þ 1Þ dw ¼

Z 1

N

x s dx ¼

1 ; s 1

ReðsÞ41:

ð1:21Þ

By analytic continuation, the integral on the left-hand side equals 1=ðs 1Þ for ReðsÞo1 as well. Now for ReðsÞo1; the function w s is also integrable over ð0; 1Þ; the result being 1=ð1 sÞ: Using the difference equation (1.16), we therefore obtain (1.20). At this point we would like to mention that the integral (1.20) was recently obtained by Broughan [4]. Likewise, for ReðsÞo0; (1.20) occurs (among many other new integrals) in a recent paper by Espinosa and Moll [7].

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Next, we deduce from the Cauchy integral formula that we may interchange the sderivative of the left-hand side of (1.20) with the integration, obtaining in particular Z

1

H 0 ð0; wÞ dw ¼ 0;

H 0 ðs; wÞ :¼ @s Hðs; wÞ:

ð1:22Þ

0

Recalling the well-known relation (Lerch formula) pﬃﬃﬃﬃﬃﬃ

H 0 ð0; wÞ ¼ log GðwÞ= 2p ;

ð1:23Þ

(cf. for example [1, p. 17]), we can rewrite (1.22) as the integral Z

1

log GðxÞ dx ¼ log

pﬃﬃﬃﬃﬃﬃ 2p:

ð1:24Þ

0

This integral is known as Raabe’s formula [9, p. 89]. Accordingly, we may view (1.19) as a generalized Raabe formula. In the case of Barnes’ multiple gamma function, (1.19) can be explicitly written out as Z 0¼ log GN ða1 x1 þ a2 x2 þ ? þ aN xn ja1 ; y; aN Þ dx: ð1:25Þ IN

(Here and from now on, we write GN;1 as GN ; likewise, zN;1 will be written zN :) Although Barnes [3, Section 53] proved a Raabe-type formula for GN ; it is complicated and involves a one-dimensional integral. (Note also that Barnes used a different normalization for his multiple gamma function.) Even in the Barnes case, formula (1.25) seems to be new. In order to give an interpretation to the integral Z zN;n ðs; W ðxÞÞ dx ¼ 0; ReðsÞoN=n ð1:26Þ IN

(which amounts to (1.4), cf. (1.6)), we observe that as x varies over I N ; the function W ðxÞ ranges over the ‘‘period parallelogram’’ PCCn ; deﬁned as the convex span of the ai : For RNpn the ai are (generically) linearly independent, so (1.26) can be restated as P zN;n ðs; wÞ dw ¼ 0; where dw is N-dimensional Lebesgue measure on the subspace of Cn spanned by the ai : This can be regarded as a ‘‘vanishing period integral’’ normalization, which ﬁxes the constant left undetermined by the difference equation (1.7). Likewise, integral (1.19) ﬁxes the constant in the difference equation (1.15). Elaborating slightly, we note that the ambiguity in the solutions of the ﬁrst-order partial analytic difference equations (1.7) and (1.15) (with the right-hand sides viewed as given functions) is not just a constant. Indeed, we can clearly add to solutions of (1.7) any meromorphic function aðwÞ having period aN ; likewise, we can multiply solutions of (1.15) by meromorphic functions mðwÞ with period aN : In the Barnes case n ¼ 1; the multiple zeta and gamma functions can be singled out by

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‘‘minimality’’: the singularities of zN and GN are enforced by the difference equation, and their asymptotics in a suitable strip is ‘‘best possible’’ [10–12]. In the Shintani case n41; however, we are dealing with partial difference equations, for which no theory of minimal solutions is known to date. In Section 4 we obtain not only integrals (1.26) and(1.19), but also generalizations to x-derivatives. Speciﬁcally, if J ¼ ðJ1 ; J2 ; y; JN Þ is a multi-index of weight jJj ¼ PN J k¼1 Jk and @x denotes the differential operator @xJ :¼

@ jJj @x1J1 ?@xNJN

;

then we show Z IN

@xJ zN;n ðs; W ðxÞÞ dx ¼ 0;

ReðsÞoðN jJjÞ=n;

ð1:27Þ

and Z IN

@xJ log GN;n ðW ðxÞÞ dx ¼ 0;

jJjoN:

ð1:28Þ

In Section 5 we show that the latter integrals easily lead to Shintani’s [13, p. 396] result stating that for kAN; the polynomial zN;n ð k; W ðxÞÞ is a sum of products of Bernoulli polynomials. We show that this also holds for the residues at each of the poles of zN;n ðs; W ðxÞÞ: It is expedient to summarize next the salient features of the Bernoulli polynomials Bl ðtÞ: We ﬁrst recall that they can be deﬁned via the generating function N X uetu Bl ðtÞ l u: ¼ eu 1 l¼0 l!

ð1:29Þ

A more instructive deﬁnition is that they are the polynomials uniquely determined by the difference equation Bl ðt þ 1Þ Bl ðtÞ ¼ ltl 1 ;

ð1:30Þ

together with the side conditions B0 ðtÞ ¼ 1;

Z

1

Bl ðtÞ dt ¼ 0;

l40:

ð1:31Þ

0

(Indeed, these relations are easily derived from (1.29).) Another important feature, namely B0l ðtÞ ¼ lBl 1 ðtÞ; is also clear from (1.29).

ð1:32Þ

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Specializing Section 5 to the Hurwitz case, we need only consider Hð k; wÞ for kAN: We have already established that Hð k; wÞ is a polynomial. Comparing the difference equations (1.17) and (1.30), we see that Hð k; wÞ coincides with Bkþ1 ðwÞ=ðk þ 1Þ up to a constant. Comparing next (1.31) and (1.20) for s ¼ k; we deduce ð1:33Þ Hð k; wÞ ¼ Bkþ1 ðwÞ=ðk þ 1Þ; kAN: Once again, this is only one of various ways to obtain this relation, which has been known for a long time. We have spelled it out, since it illustrates our approach to the general case in Section 5. In Section 5 we also prove three identities arising for s ¼ 0; namely n 1X z ð0; wj ja1j ; y; aNj Þ; zN;n ð0; wja1 ; y; aN Þ ¼ n j¼1 N zN;n ð0; wja1 ; y; aN Þ ¼ ð 1ÞN zN;n ð0; A wja1 ; y; aN Þ; Nþ1

GN;n ðwÞðGN;n ðA wÞÞð 1Þ

¼

n Y

ðA :¼ a1 þ ? þ aN Þ Nþ1

GN ðwj ÞðGN ðAj wj ÞÞð 1Þ

;

j¼1

where GN;n ðwÞ ¼ GN;n ðwja1 ; y; aN Þ;

GN ðwj Þ ¼ GN ðwj ja1j ; y; aNj Þ:

Furthermore, we determine the polynomial zN;n ð0; wÞ explicitly. In essence, all of these s ¼ 0 results were obtained ﬁrst by Shintani [16, pp. 206, 210]. We begin Section 6 by detailing a slight generalization of our assumptions, for which all of the previous results still hold. Speciﬁcally, we allow the numbers a1j ; y; aNj to lie in any half-plane obtained by rotating the right half-plane over an angle less than p=2: (This angle restriction prevents multi-valuedness.) It is clear from the Dirichlet series (1.5) that we can do this when we choose wj in the same half-plane, but a complete account of the pertinent analytic continuation involves a little more effort. This generalization—already present in Barnes’ and Shintani’s work—enables us to relate Barnes’ GN to certain inﬁnite products. The integral formulas (1.26) and (1.19) (with n ¼ 1) allow us to make this relation completely explicit. The pertinent result (Proposition 6.1) reduces to the reﬂection equation for G1 ; whereas for G2 it amounts to a result that can be found in Barnes’ and Shintani’s papers. For GN with N42; Proposition 6.1 seems to be new. As a corollary, we ﬁnd an explicit relation between G3 and the elliptic gamma function introduced in [10]. Since we are promoting in this paper a simpliﬁed approach to multiple zeta and gamma functions, we do not assume any familiarity with them. Although the theory of minimal solutions of ﬁrst-order analytic difference equations [10] provided an important motivation for our work, we make no further appeal to this theory. Likewise, we avoid any reference to number theory, although this was the main motivation for Shintani’s work. Lastly, we would like to mention that other approaches to special values of zN;n ðs; wÞ can be found in [5,6].

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2. Analytic continuation of fN;n The meromorphic continuation of zðs; M; xÞ was proved by Shintani [13], who was interested in using it to calculate special values. Had he not been interested in this, he could have deduced the meromorphic continuation of zðs; M; xÞ from an old result of Mahler’s [8, Section 19]. We shall actually consider a generalization ZN;n ðS; w; MÞ of zN;n ðs; wÞ (with M deﬁned by (1.1)), replacing s by n complex variables S ¼ ðS1 ; y; Sn Þ in the half-space Pn j¼1 ReðSj Þ4N: Namely, ZN;n ðS; w; MÞ :¼

N X

n Y

ðwj þ m1 a1j þ ? þ mN aNj Þ Sj :

ð2:1Þ

m1 ;y;mN ¼0 j¼1

Since we are assuming Reðaij Þ40 and Reðwj Þ40; we may and will choose the principal branch of the logarithm to deﬁne the complex powers in (2.1). To determine the region of absolute convergence of (2.1), let c :¼ minfReðaij Þg40; i;j

C :¼ 1 þ max fjaij jg: i;j

ð2:2Þ

Then, for mi X0; CNðjjwjj þ jjmjjÞXjwj þ m1 a1j þ ? þ mN aNj j XReðwj þ m1 a1j þ ? þ mN aNj Þ4c

N X

mi Xc jjmjj;

ð2:3Þ

i¼1

P Pn 2 2 2 n N where jjmjj2 :¼ N i¼1 jmi j ; jjwjj :¼ j¼1 jwj j for mAR ; wAC : Thus (2.1) conPn verges absolutely if and only if j¼1 ReðSj Þ4N: From this it readily follows that it deﬁnes an analytic function for ðS; w; MÞ in the subset of Cn Dn DN;n given by Pn j¼1 ReðSj Þ4N; with Dn and DN;n given by (1.12) and (1.13). The zeta function ZN;n ðS; w; MÞ reduces to zN;n ðs; wja1 ; y; aN Þ when all the Sj are equal to s and ReðsÞ4N=n: Furthermore, for ReðsÞ4N; ZN;n ððs; 0; 0y; 0Þ; w; MÞ ¼ zN ðs; w1 ja11 ; y; aN1 Þ:

ð2:4Þ

Similarly, by restriction of the S-variable, ZN;n yields any zN;n0 with n0 pn: However, ZN;n has the drawback of being singular at S ¼ 0; as we shall see in Section 3. To keep this paper as self-contained as possible, rather than rely on Mahler’s paper, we now give a detailed proof of the meromorphic continuation of zN;n : However, in later sections we shall need no formulas affording it. We will only use holomorphy of the function zN;n ðs; wja1 ; y; aN Þ=pN;n ðsÞ in the domain C Dn DN;n ; as already discussed in the Introduction. Readers who are willing to take this analyticity for granted can safely pass to Proposition 2.2.

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To establish the analytic continuation of ZN;n ðS; w; MÞ; we ﬁrst use Euler’s formula (1.9) to obtain the integral representation Z n n n Y Y X S 1 GðSj Þ ¼ hðtÞ tj j dt; ReðSj Þ40; ReðSj Þ4N; ZN;n ðS; w; MÞ tARnþ

j¼1

j¼1

j¼1

ð2:5Þ where Rnþ :¼ ð0; NÞn CRn ; dt is Lebesgue measure on Rn ; and hðtÞ :¼

N X

n Y

exp tj ðwj þ m1 a1j þ ?mN aNj Þ

m1 ;y;mN ¼0 j¼1

Qn wj tj j¼1 e

f ðtÞ; ¼ Q P N n i¼1 j¼1 aij tj with f ðtÞ :¼

YN i¼1

j

!

n X

aij tj ;

jðrÞ :¼

j¼1

r : 1 e r

ð2:6Þ

Following Shintani [13, Section 1], we write Rnþ

¼ A1 ,A2 ,?,An ;

Aj :¼

tARnþ tj

¼ max ftk g ; 1pkpn

where the union is disjoint up to sets of measure 0. Then (2.5) becomes n n n Y Xn Z Y X Sk 1 ZN;n ðS; w; MÞ GðSj Þ ¼ hðtÞ t dt ¼: Ij ðS; w; MÞ: k j¼1 j¼1

Aj

ð2:7Þ

ð2:8Þ

j¼1

k¼1

For tAAj ; we switch to new coordinates ðr; sÞ; where r :¼ tj and sk :¼ tk =tj ð1pkpn; kajÞ: The Jacobian determinant is rn 1 ; and the new coordinates range over sk Að0; 1Þ; rAð0; NÞ: For convenience, on Aj we deﬁne sj :¼ 1; s ¼ ðs1 ; y; sn Þ: We now change to the new coordinates. The piece of the integral (2.8) corresponding to j ¼ 1 becomes Z 1 Z N Z 1 n Pn Y N 1þ Sk rw1 k¼1 I1 ðS; w; MÞ ¼ r e ? gðr; sÞ sSk k 1 dsk dr; ð2:9Þ r¼0

s2 ¼0

sn ¼0

k¼2

where Q f ðr; rs2 ; y; rsn Þ nk¼2 e rsk wk gðr; sÞ :¼ : Pn QN i¼1 ðai1 þ k¼2 aik sk Þ

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For the innermost integral in (2.9), we integrate by parts to get Z

1

sn ¼0

snSn 1 gðr; sÞ dsn

1 ¼ Sn ¼

1 Sn

Z Z

1

sSn n

sn ¼0 1

sn ¼0

@g ðr; sÞ ðSn þ 1Þgðr; s2 ; y; sn 1 ; 1Þ dsn @sn

sSn n g0 ðS; r; sÞ dsn ;

ð2:10Þ

with the obvious deﬁnition of g0 : We can repeat the integration by parts M times in (2.10) to get !Z Z 1 M 1 Y 1 sSn n 1 gðr; sÞ dsn ¼ snSn þM gM ðS; r; sÞ dsn ; S þp sn ¼0 sn ¼0 p¼0 n where gM is a sum of sn -derivatives of g (and some specializations of them at sn ¼ 1) with coefﬁcients which are monomials in Sn : We have thus replaced the exponent sSn n 1 in (2.7) by sSn n þM : The same procedure, applied to the remaining sk in (2.9), yields I1 ðS; w; MÞ ¼ TM;1 ðSÞ

Z

Pn

N

r

k¼1

Sk N 1 rw1

Z gðS; ˜ r; sÞ

e

r¼0

s

n Y

skSk þM dsk dr;

k¼2

ð2:11Þ where TM;j ðSÞ :¼

Y

M Y

1pkpn p¼0 kaj

1 ; Sk þ p

Z

Z :¼ s

Z

1

1

;

? s2 ¼0

ð2:12Þ

sn ¼0

and where gðS; ˜ r; sÞ ¼

X

hr ðSÞfr ðr; s2 ; y; sn Þ

r

is again essentially a ﬁnite sum of sk -derivatives fr of g; with coefﬁcients hr which are monomials in the Sk : We now expand each fr in powers of r to obtain fr ðr; sÞ ¼ rM hr;M ðr; sÞ þ

M 1 X

bl;r ðsÞrl ;

l¼0

with jhr;M ðr; sÞj bounded above by a polynomial in r for rX0:

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Returning to our integral (2.11), we ﬁnd 0 M 1 n X X Gð N þ l þ Pn Sk Þ Z Y Pn k¼1 hr ðSÞ bl;r ðsÞ skSk þM dsk I1 ¼ TM;1 ðSÞ@ Nþlþ S k s k¼1 r l¼0 k¼2 w1 ! Z Z n Pn N X Y M N 1þ S S þM k¼1 k h þ hr ðSÞ e rw1 r sk k dsk dr : r;M ðr; sÞ r

r¼0

s

k¼2

ð2:13Þ We have proved most of Proposition 2.1. The function ZN;n ðS; w; MÞ=Gð N þ morphic function on Cn Dn DN;n : The function

Pn

j¼1

Sj Þ extends to a holo-

zN;n ðs; wja1 ; y; aN ÞGðsÞ=Gðns NÞ extends to a holomorphic function on C Dn DN;n : In particular, for a fixed ðw; MÞADn DN;n ; the function s/zN;n ðs; wja1 ; y; aN Þ is meromorphic; it has at most simple poles for s ¼ ðN lÞ=n with lAN; and has no poles for s ¼ k with kAN: As remarked earlier, the meromorphic continuation of zN;n ðs; M; xÞ was found by Shintani [13, Section 1]. He did not explicitly locate the poles, but they are easily deduced from his formulas. Proof. Deﬁnition (2.12) of TM;j ðSÞ entails that the functions Q

TM;j ðSÞ ; GðSk Þ

1pjpn;

ð2:14Þ

1pkpn kaj

are entire. Hence we deduce from (2.13) and its analogs for I2 ; y; In that the functions Gð N þ

Ij ðS; w; MÞ P Q ; n k¼1 Sk Þ kaj GðSk Þ

j ¼ 1; y; n;

extend holomorphically to the domains given by ReðSk Þ4 M 1; kaj;

n X

ReðSk Þ4 M þ N; ðw; MÞADn DN;n :

k¼1

As M is arbitrary, the ﬁrst assertion of the proposition now readily follows from (2.8). Taking Sj ¼ s for all j; we obtain the second one. & Since s ¼ 0 is a regular point of zN;n ðs; wja1 ; y; aN Þ; we can deﬁne a multiple gamma function by (1.14). Then the gamma recurrence (1.15) easily follows from the

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zeta recurrence (1.7). From its deﬁnition we see that GN;n ðwja1 ; y; aN Þ is holomorphic on Dn DN;n : We now use (1.15) to show that GN;n ðwja1 ; y; aN Þ continues meromorphically to Cn DN;n : Proposition 2.2. The function 1=GN;n ðwj a1 ; y; aN Þ extends to a holomorphic function on Cn DN;n : Its zero locus consists of the hyperplanes Hj :¼ fwACn j wj ¼ 0g and their translates Hj ðm1 a1 þ ? þ mN aN Þ; with j ¼ 1; y; n and m1 ; y; mN AN: Although Shintani did not explicitly consider the nature of GN;n ðwÞ; Proposition 2.2 for w ¼ W ðxÞ (cf. (1.6)) is a direct consequence of Barnes’ study of GN and of Shintani’s formula [16, p. 204] relating GN;n to GN (see Proposition 3.2 below). Proof. Taking N ¼ 1; we iterate the difference equation (1.15), obtaining ! l 1 Y n Y 1 1 ¼ ; lANþ : ðwj þ ma1j Þ G1;n ðwj a1 Þ G1;n ðw þ la1 j a1 Þ m¼0 j¼1 From this we deduce that 1=G1;n ðwj a1 Þ extends holomorphically to ðDn la1 Þ D1;n : Since Reða1j Þ40 for j ¼ 1; y; n; and l is arbitrary, it follows that 1=G1;n ðwja1 Þ extends to a holomorphic function on Cn D1;n ; whose zero locus equals Hj ma1 with j ¼ 1; y; n and mAN: Using induction on N; we now obtain the proposition. &

3. Degree-m polynomials at s ¼ ðN mÞ=n s s The ladder of difference equations (1.7) begins with z0;n ðs; wÞ ¼ w s 1 w2 ?wn ; which is evidently a polynomial of degree kn when s equals an integer kAN: Although we know of no theory of minimal solutions to ﬁrst-order partial difference equations in Cn ; it is natural to surmise that the Nth level of the ladder zN;n ð k; wÞ is a polynomial of degree N more than that of the base level. As a ﬁrst step, we prove the polynomial property by showing that the pertinent wderivatives of zN;n ðs; wÞ vanish identically at these s-values. Speciﬁcally, for a multiP index J ¼ ðJ1 ; y; Jn Þ of weight jJj ¼ nj¼1 Jj ; denote by @wJ the differential operator @ jJj : J @w1 1 ?@wnJn

In the region of absolute convergence of the Dirichlet series, direct

differentiation yields @wJ zN;n ðs; wÞ

¼ ð 1Þ

jJj

Jj 1 n Y Y j¼1

p¼0

! ðs þ pÞ

n X Y mANN

ðwj þ m1 a1j þ ? þ mN aNj Þ s Jj

j¼1

¼ ð 1ÞjJj ZN;n ððs þ J1 ; y; s þ Jn Þ; w; MÞ

Jj 1 n Y Y ðs þ pÞ; j¼1

p¼0

ð3:1Þ

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P where ZN;n was deﬁned in (2.1). The above series converges for No nj¼1 Reðs þ Jj Þ ¼ jJj þ n ReðsÞ: In particular, by analytic continuation in s; series (3.1) represents @wJ zN;n ð k; wÞ for jJj4N þ kn: Just as in the example of the Hurwitz zeta function in Section 1, the analyticity of @wJ zN;n ðs; wÞ for ReðsÞ4ðN jJjÞ=n entails that the residues at the poles in this sregion have been differentiated away. Therefore, the residues at the poles ðN mÞ=n with mAN are polynomials of degree at most m: For m of the form N þ kn with kAN; we showed in Proposition 2.1 that there are no poles. Since the product term in (3.1) vanishes for s ¼ k and jJj4N þ kn; we infer that zN;n ð k; wÞ is a polynomial of degree at most N þ kn: We now extend these results. Proposition 3.1. The functions PknþN;N;n ðwÞ :¼ zN;n ð k; wÞ;

kAN;

ð3:2Þ

are polynomials of degree kn þ N: For NX1; zN;n ðs; wÞ has simple poles at s ¼ sl;N;n ; where sl;N;n :¼ ðN lÞ=n; lAN\ðN þ nNÞ;

ð3:3Þ

with residues Pl;N;n ðwÞ that are polynomials of degree l; except possibly for non-generic MADN;n : More precisely, whenever sl;N;n p1=n; the degree equals l on all of DN;n ; in particular, the degree of the polynomials Pm;1;n ðwj aÞ equals m for all mAN and p Þ; aAD1;n : For N41 and sl;N;n 41=n; the degree equals l on the polysector SN;n ð2ðN lÞ where SN;n ðfÞ :¼ fMADN;n j jArgðaij Þjofg;

fAð0; p=2Þ:

ð3:4Þ

All the above statements concerning the regular values were proved by Shintani [13, Section 1] in the case w ¼ W ðxÞ (cf. (1.6)). Proof. For the regular values s ¼ k; it remains to show that the degree of the polynomials (3.2) equals the upper bound kn þ N already established above. We prove this via the difference equations (1.7), as follows. Let us assume that the polynomial (3.2) has degree Lokn þ N: Now consider the monomials of highest degree L occurring in the two polynomials on the lhs of PknþN;N;n ðw þ aN ja1 ; y; aN Þ PknþN;N;n ðwja1 ; y; aN Þ ¼ PknþN 1;N 1;n ðwja1 ; y; aN 1 Þ:

ð3:5Þ

Clearly, their differences yield terms whose degree is at most L 1: Thus the degree of the polynomial on the right-hand side is at most L 1; too. Repeating this argument, we deduce that Pkn;0;n ðwÞ has degree at most kn 1; contradicting z0;n ð k; wÞ ¼ wk1 ?wkn :

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Passing to the pole and residue assertions, we ﬁrst study the case that the numbers wj and aij are positive. Consider the behavior of series (1.5) for real s near s0;N;n : It follows from the paragraph containing estimate (2.3) that it diverges as sks0;N;n : Thus there must be a pole at s ¼ s0;N;n ; yielding a constant non-zero residue. Turning to s1;N;n (for N41), we inspect (3.1) with jJj ¼ 1: As before, we get a divergence for sks1;N;n ; so s1;N;n is a pole with residue a degree-1 polynomial. Clearly, this reasoning can be repeated for s ¼ sl;N;n ; so for positive aij the degree is always l: By analyticity in DN;n ; the degree is therefore generically equal to l on DN;n : To obtain the stronger assertions concerning the degree, we ﬁrst take N ¼ 1: Now we reconsider series (1.5), ﬁxing wAð0; NÞn and aAD1;n : Since a is ﬁxed, it belongs to a sector S1;n ðfÞ for some fop=2: As we let sks0;N;n ¼ 1=n; all of the terms in the series belong to the sector S1;n ðf0 Þ with f0 Aðf; p=2Þ for s sufﬁciently close to s0;N;n : Then the real parts of the terms in the series are bounded below by cosðf0 Þ times their modulus, so divergence as sks0;N;n follows as before from (2.3). For sl;N;n we apply this argument to series (3.1) with jJj ¼ l; obtaining once more divergence as sksl;N;n : We have therefore proved the degree assertion for N ¼ 1: Letting now N41 and sl;N;n p1=n (so that lXN 1), we can use the difference equations (1.7) in the same way as before to obtain the degree l assertion. To be quite speciﬁc, we can multiply (1.7) by ðs sl;N;n Þ and take s to sl;N;n to get Pl;N;n ðw þ aN ja1 ; y; aN Þ Pl;N;n ðwja1 ; y; aN Þ ¼ Pl 1;N 1;n ðwja1 ; y; aN 1 Þ:

ð3:6Þ

Iterating downward, we can relate Pl;N;n to the degree-ðl N þ 1Þ polynomial Pl Nþ1;1;n : It remains to prove the last assertion. Taking ﬁrst l ¼ 0; we need only inspect the argument variation of the terms in the pertinent series to obtain the desired divergence for M in the speciﬁed sector. For l40 we can use (3.6) once more, this time to relate Pl;N;n to the non-zero constant P0;N l;n : & In fact, we surmise that the non-generic subsets of DN;n where the degree is lower than l are empty. We proceed by pointing out that (3.1) yields in particular @n z ðs; wÞ ¼ ð sÞn zN;n ðs þ 1; wÞ: @w1 @w2 ?@wn N;n

ð3:7Þ

For s ¼ k with kANþ ; this may be viewed as a generalization of the Bernoulli property (1.32). Specializing to the Barnes case n ¼ 1; it entails that the above nongeneric subsets of DN;1 are indeed empty. To explain this, we recall that zN ð0; wÞ has degree N; so (3.7) implies that the residue at s ¼ 1 has degree N 1; etc. (Alternatively, the explicit residue formulas in terms of Barnes’ multiple Bernoulli polynomials can be invoked, cf. Eq. (3.9) in [11].) Unfortunately, for n41 the partial differential operator occurring in (3.7) can lower the degree of polynomials by more than n; so that it cannot be used to rule out non-generic degree lowering.

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We can use Proposition 3.1 to verify that ZN;n ðS; w; MÞ is singular at S ¼ 0 for N40 and n41; as mentioned in Section 2. Indeed, if the origin were regular, it would follow from (2.4) that ZN;n ð0; w; MÞ equals zN ð0; w1 ja11 ; y; aN1 Þ (by taking s-0). Likewise, ZN;n ð0; w; MÞ would be equal to zN ð0; wj ja1j ; y; aNj Þ for 1ojpn: This would imply that zN ð0; w1 ja11 ; y; aN1 Þ is constant as a function of w1 ; contradicting Proposition 3.1. Turning to the multiple gamma function GN;n ðwÞ (1.14), we notice that at the base level N ¼ 0 of the ladder (1.15) we have n X log G0;1 ðwj Þ: log G0;n ðwÞ ¼ j¼1

As before, we expect the same to hold for arbitrary N; up to a polynomial of degree at most N: To study this, we note that for jJjXN þ 1; the series in (3.1) converges for ReðsÞ4 1=n: We can differentiate it with respect to s and set s ¼ 0 to obtain @s @wJ zN;n ðs; wja1 ; y; aN Þjs¼0 ( ð 1ÞjJj GðjJjÞzN ðjJj; wj ja1j ; y; aNj Þ; if jJj ¼ Jj for some j; ¼ 0 otherwise: Applying the above to zN;n and to zN;1 ¼ zN ; we obtain for all jJjXN þ 1; ! n X J 0 ¼ @w @s zN;n ðs; wja1 ; y; aN Þjs¼0 @s zN ðs; wj ja1j ; y; aNj Þjs¼0 : j¼1

Recalling deﬁnition (1.14), we see that GN;n reduces to a sum of Barnes’ GN functions, up to a polynomial of degree at most N: We now render this result more precise. To this end we deﬁne for aADn a coefﬁcient vector cðaÞACn by cðaÞj :¼

n 1 X ðlogðak Þ logðaj ÞÞ; naj k¼1

j ¼ 1; y; n;

ð3:8Þ

the logarithm branch being the principal one. Proposition 3.2. Let NX1 and nX2: Then we have log GN;n ðwja1 ; y; aN Þ ¼ PN;n ðwja1 ; y; aN Þ þ

n X

log GN ðwj ja1j ; y; aNj Þ;

ð3:9Þ

j¼1

with PN;n ðwÞ a polynomial of degree at most N: Moreover, P1;n ðwÞ is given by P1;n ðwjaÞ ¼

n X j¼1

wj cðaÞj ;

ð3:10Þ

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and the degree of PN;n ðwja1 ; y; aN Þ equals N whenever at least one of a1 ; y; aN satisfies cðai Þa0: For w ¼ W ðxÞ; the polynomial in (3.9) was made quite explicit by Shintani [16, pp. 204, 206]. We return to his formula below (5.10). Proof. Thanks to the difference equations PN;n ðw þ aN ja1 ; y; aN Þ PN;n ðwja1 ; y; aN Þ ¼ PN 1;n ðwja1 ; y; aN 1 Þ;

ð3:11Þ

we need only show (3.10). Indeed, we have already proved that PN;n ðwÞ is a polynomial of degree at most N; and (3.10) shows that P1;n ðwjaÞ has degree 1, provided cðaÞa0: Hence the degree assertion follows from (3.11) by the argument in the paragraph containing (3.5). By analyticity in w; it sufﬁces to prove (3.10) for wAð0; NÞn ; which we require from now on. We begin by observing that for ReðsÞ41=n we can invoke the Dirichlet series representation to obtain z1;n ðs; wjaÞ ¼

n Y

! a s j

z1;n ðs; ðw1 =a1 ; y; wn =an Þj 1Þ;

1j :¼ 1; j ¼ 1; y; n:

ð3:12Þ

j¼1

Likewise, for ReðsÞ41 we have z1;1 ðs; wj j aj Þ ¼ a s j Hðs; wj =aj Þ;

j ¼ 1; y; n;

ð3:13Þ

with Hðs; wÞ the Hurwitz zeta function (1.8). Using meromorphic continuation in s and regularity at s ¼ 0; this yields P1;n ðwjaÞ ¼

n X

logðaj Þ½z1;n ð0; ðw1 =a1 ; y; wn =an Þj 1Þ Hð0; wj =aj Þ

j¼1

þ @s z1;n ðs; ðw1 =a1 ; y; wn =an Þj 1Þjs¼0

n X

H 0 ð0; wj =aj Þ:

ð3:14Þ

j¼1

Next, we study the difference function z1;n ðs; xj 1Þ

n 1X Hðns; xj Þ ¼: dn ðs; xÞ; n j¼1

xA½1; NÞN :

ð3:15Þ

From the series representation we have dn ðs; xÞ ¼

N X m¼0

In ðm; s; xÞ;

ReðsÞ41=n;

ð3:16Þ

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where In ðm; s; xÞ :¼

n Y

ðxj þ mÞ s

j¼1

n 1X ðxj þ mÞ ns : n j¼1

ð3:17Þ

Here, xA½1; NÞN is viewed as ﬁxed, whereas m varies over N and s over C; with the principal branch of the logarithm understood for the complex powers. Clearly, this function is entire in s and obeys the bound In ðm; s; xÞ ¼ Oðm ns 2 Þ;

ð3:18Þ

m-N:

Hence we infer that the series on the rhs of (3.16) converges for ReðsÞ4 1=n: By analytic continuation, it follows that (3.16) actually holds for ReðsÞ4 1=n: Since In ðm; s; xÞ vanishes at s ¼ 0; we obtain from (3.15) z1;n ð0; xj 1Þ ¼

n n 1X 1 1X Hð0; xj Þ ¼ xj ; n j¼1 2 n j¼1

ð3:19Þ

where we used (1.33). By analyticity in s; we may interchange the s-derivative of (3.16) with the summation, so we also have N X @s dn ðs; xÞ ¼ @s In ðm; s; xÞ; ReðsÞ4 1=n: ð3:20Þ m¼0

Now the s-derivative of In vanishes at s ¼ 0 as well, so we obtain from (3.15) n X H 0 ð0; xj Þ: ð3:21Þ @s z1;n ðs; xj 1Þjs¼0 ¼ j¼1

We continue by noting that the functions appearing in (3.19) and (3.21) are analytic for xADn ; so that we may substitute xj -wj =aj : Substitution in (3.14) then yields ! ! n n X 1 1X wk 1 wj P1;n ðwjaÞ ¼ logðaj Þ : 2 n k¼1 ak 2 aj j¼1 This can be rewritten as (3.10), so our proof is complete.

&

We have gone to some lengths to obtain the explicit formula (3.10), since it has two illuminating features (apart from implying the degree-N property). First, consider the difference equation (3.11) with N ¼ 1: Obviously, the right-hand side vanishes, so if we were dealing with an ordinary difference equation, it would follow that P1;n ðwÞ could be at most a non-zero constant. That P1;n ðwÞ can have degree 1 for the partial difference equation at issue is due to the existence in Cn ; n41; of a coefﬁcient vector cðaÞ (see (3.8)) that is orthogonal to a in the sense that n X j¼1

aj cðaÞj ¼ 0:

ð3:22Þ

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Secondly, when we specialize P1;n ðwjaÞ to the Shintani case, where w is replaced by W ðxÞ ¼ xa; the polynomial vanishes identically by (3.22). Another interesting corollary of our calculation leading to (3.10) is an explicit formula for the residue of z1;n ðs; wjaÞ at its simple pole s ¼ 1=n; namely, n 1 1Y 1=n a : ð3:23Þ lim s z1;n ðs; wjaÞ ¼ n n j¼1 j s-1=n Indeed, the residue of Hðs; wÞ at s ¼ 1 equals 1 (as is for example plain from (1.11) with M ¼ 1), so from (3.15) and analyticity of dn ðs; xÞ for ReðsÞ4 1=n we obtain (3.23) for a ¼ 1: The general case is then clear from (3.12).

4. Vanishing zeta and gamma integrals In this section we obtain the ‘‘vanishing period integrals’’ (1.26)–(1.27) and their corollary (1.28). Proposition 4.1. Let NANþ ; ReðsÞoN=n and s not a pole of zN;n ðs; wÞ: Then the function x/zN;n ðs; W ðxÞÞ is integrable with respect to Lebesgue measure dx on the unit N-dimensional cube I N :¼ ð0; 1ÞN ; and satisfies Z zN;n ðs; W ðxÞÞ dx ¼ 0; ReðsÞoN=n: ð4:1Þ IN

More generally, x/@xJ zN;n ðs; W ðxÞÞ is integrable on I N for ReðsÞoðN jJjÞ=n; and satisfies Z @xJ zN;n ðs; W ðxÞÞ dx ¼ 0; ReðsÞoðN jJjÞ=n: ð4:2Þ IN

Finally, we have the integrals Z @xJ Pm;N;n ðW ðxÞÞ dx ¼ 0;

0pjJjom; mANþ ;

ð4:3Þ

IN

where Pm;N;n ðwÞ are the polynomials from Proposition 3.1. Corollary 4.2. The function log GN;n ðW ðxÞÞ is integrable on I N ; and satisfies Z log GN;n ðW ðxÞÞ dx ¼ 0:

ð4:4Þ

IN

More generally, for N41 and jJjoN the function @xJ log GN;n ðW ðxÞÞ is integrable on I N ; and satisfies Z @xJ log GN;n ðW ðxÞÞ dx ¼ 0; 0pjJjoN: ð4:5Þ IN

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Proof of Proposition 4.1. Just as in the Hurwitz case treated in Section 1, we subtract the m ¼ 0 term from the Dirichlet series (1.5). This yields a function n Y bzN;n ðs; wÞ :¼ zN;n ðs; wÞ w s ð4:6Þ j ; j¼1

whose analyticity domain in the wj is larger than Dn ; containing in particular the origin, where the m ¼ 0 term is singular. Speciﬁcally, an inspection of the series shows that for ReðsÞ4N=n we have holomorphy in Vn :¼ fwACn j Reðwj Þ4 c; 1pjpng; with c40 given by (2.2) (say). To see that this actually holds true whenever s is not a pole of zN;n ðs; wÞ; we note that the difference equation (1.7) yields bzN;n ðs; wÞ ¼ zN;n ðs; w þ aN Þ þ bzN 1;n ðs; wÞ;

bz0;n ðs; wÞ ¼ 0:

By induction on N; this entails that bzN;n ðs; wÞ is indeed holomorphic in Vn : Thus, for any value of s outside the poles, bzN;n ðs; W ðxÞÞ is inﬁnitely differentiable in x on an open neighborhood of ½0; 1 N : The map x/bzN;n ðs; W ðxÞÞ; as well as any of its x-derivatives, is therefore integrable on I N ¼ ð0; 1ÞN ; so the function Z b J ðsÞ :¼ @xJ bzN;n ðs; W ðxÞÞ dx Q

ð4:7Þ

IN

is well deﬁned and analytic in s away from the poles of zN;n ðs; wÞ: Now from (4.6) we have for xAI N zN;n ðs; W ðxÞÞ ¼ bzN;n ðs; W ðxÞÞ þ W ðxÞ SðsÞ ;

ð4:8Þ

where wS :¼

n Y

S

wj j ;

SðsÞ :¼ ðs; s; y; sÞACn ;

wADn :

j¼1

It follows that @xJ zN;n ðs; W ðxÞÞ is integrable on I N if and only if @xJ W ðxÞ SðsÞ is. (We shall see shortly that this happens if ReðsÞoðN jJjÞ=n:) b J ðsÞ; assuming ﬁrst that s belongs to the region We proceed by computing Q ReðsÞ4N=n; where the Dirichlet series and all series obtained by taking term-wise derivatives converge absolutely. Then we have from (4.6) and (4.7) Z n X0 Y b J ðsÞ ¼ Q @xJ ðW ðxÞj þ m1 a1j þ ? þ mN aNj Þ s dx IN

¼

Z

IN

m

@xJ

X0 m

j¼1

ðW ðx þ mÞÞ

SðsÞ

dx ¼

Z N RN þ I

@xJ W ðxÞ SðsÞ dx;

ð4:9Þ

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where the sums are over all m ¼ ðm1 ; y; mN Þ; save for the term m ¼ 0: By induction on jJj; we see that the integrand is of the form n X Y @xJ W ðxÞ SðsÞ ¼ cIJ ðsÞ ðx1 a1j þ ? þ xN aNj Þ s Ij I

¼

X

j¼1

cIJ ðsÞ W ðxÞ SðsÞ I ;

ð4:10Þ

I

where I ¼ ðI1 ; y; In Þ ranges over multi-indices satisfying jIj ¼ jJj; and cIJ ðsÞ is a polynomial in s whose coefﬁcients depend on J and on the aij ; but not on x: We now switch to cubical coordinates ðr; sÞ on RN ; as we did in Section 2 (between formulas (2.8) and (2.9)) without giving them this name. Namely, we adopt coordinates with respect to the unit ‘‘sphere’’ C N 1 :¼ fsARN jjsjjN ¼ 1g; jjxjjN :¼ max fjxi jg: 1pipN

Thus, r ¼ rðxÞ :¼ jjxjjN and s ¼ sðxÞ :¼ r 1 xAC N 1 : The new volume element is rN 1 dr ds; where ds is the ðN 1Þ-dimensional Lebesgue measure on C N 1 : We let CþN 1 :¼ C N 1 -RN þ: Then we have Z N Z Z W ðxÞ SðsÞ I dx ¼ W ðsÞ SðsÞ I rN 1 ns jJj dr ds: ð4:11Þ N RN þ I

N 1 sACþ

r¼1

The crux is now that the r-integral in (4.11) is elementary, so that from (4.9) and (4.10) we get Z X 1 b J ðsÞ ¼ Q cIJ ðsÞ W ðsÞ SðsÞ I ds: ð4:12Þ N jJj ns I sACþN 1 The integrals over CþN 1 in (4.12) yield entire functions of s; and the coefﬁcients cIJ ðsÞ are polynomials in s: As a consequence, (4.12) holds for sAC: (In particular, the only b J ðsÞ occurs for s ¼ ðN jJjÞ=n:) eventual pole of Q Next, we use (4.10) and the above change of variables to verify that the function Z J P ðsÞ :¼ @xJ W ðxÞ SðsÞ dx ð4:13Þ IN

is well deﬁned and analytic for ReðsÞoðN jJjÞ=n: Indeed, we have Z Z 1 X P J ðsÞ ¼ cIJ ðsÞ W ðsÞ SðsÞ I rN 1 ns jJj dr ds; I

N 1 sACþ

ð4:14Þ

r¼0

where the s-integrand is bounded and the r-integral is again elementary. Comparing the result to (4.12), we obtain b J ðsÞ; P J ðsÞ ¼ Q

ReðsÞoðN jJjÞ=n:

ð4:15Þ

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This is the key equality: combining it with (4.8) and deﬁnitions (4.7) and (4.13) of b J ðsÞ and P J ðsÞ; resp., we obtain integral (4.2). Q Integrals (4.3) with m ¼ kn þ N; kAN; are immediate from (4.2) with s ¼ k; cf. (3.2). To obtain them for the residue polynomials, we need only divide all of the above quantities by Gðns NÞ to obtain entire functions of s: Letting s converge to the locations sl;N;n (3.3), the residue integrals (4.3) result. & To appreciate the above proof in one fell swoop, it may help to reinspect the reasoning for the Hurwitz case, cf. the paragraph containing (1.21). Key equality (4.15) can be viewed as a higher-dimensional version of the equality of the integrals RN R 1 s dx and 1 x s dx; in the sense of their analytic continuations to C\f1g 0 x being equal. Proof of Corollary 4.2. As we have seen in the above proof, the integrand in (4.7) is continuous in ðs; xÞ on sets of the form K ½0; 1 N ; where K is any s-compact not containing poles of zN;n : Thus we may interchange the s-derivative of the integral with the integration. Take jJjoN and ReðsÞo1=n from now on. Using the Cauchy integral formula and dominated convergence, we deduce from Eq. (4.14) that the sderivative of P J ðsÞ exists and that we may interchange the s-derivative of the integral on the rhs of (4.13) with the integration. Recalling (4.8), we now see that @s @xJ zN;n ðs; W ðxÞÞ is integrable on I N ; and that we have Z Z d @xJ zN;n ðs; W ðxÞÞ dx ¼ @s @xJ zN;n ðs; W ðxÞÞ dx: ð4:16Þ ds I N IN Since the lhs vanishes by (4.2), so does the rhs. Hence the assertions follow upon taking s ¼ 0 (recall (1.14)). &

5. Applications of the vanishing period integrals to Shintani–Barnes functions In this section we exploit the results of Section 4 to derive various explicit formulas, most of which were obtained before by Shintani. We shall need Lemma 5.1. Suppose RðxÞAC½x1 ; y; xN is a polynomial of degree at most m satisfying Z @xJ RðxÞ dx ¼ 0; 0pjJjom: ð5:1Þ IN

Then we have the identities Rð1 xÞ ¼ ð 1Þm RðxÞ;

ð1 xÞi :¼ 1 xi ; 1pipN;

ð5:2Þ

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and RðxÞ ¼

X

cL

L jLj¼m

BL ðxÞ ; L!

ð5:3Þ

where L ranges over all N-multi-indices of weight m; Z cL :¼ IN

@xL RðxÞ dx;

L! :¼

N Y

BL ðxÞ :¼

Li !;

i¼1

N Y

BLi ðxi Þ;

ð5:4Þ

i¼1

with Bl ðtÞ the Bernoulli polynomial defined in (1.29). Moreover, any polynomial of form (5.3) satisfies (5.1). Proof. Since RðxÞ has degree at most m; the difference polynomial DðxÞ :¼ Rð1 xÞ ð 1Þm RðxÞ has degree at most m 1: Next, we observe that (5.1) and the change of variables x/1 x imply Z @xJ DðxÞ dx ¼ 0; 0pjJjpm 1: IN

Hence it recursively follows that the coefﬁcients of the terms of degree m 1; m 2; y; 0 vanish, yielding (5.2). To prove (5.3), we ﬁrst use (1.31) and (1.32) to obtain Z IN

@xJ BL ðxÞ dx

¼

L! if L ¼ J; 0

otherwise:

ð5:5Þ

Now let cL be given by (5.4) and set QðxÞ :¼ RðxÞ

X L jLj¼m

cL

BL ðxÞ : L!

R Then (5.1) and (5.5) imply I N @xJ QðxÞ dx ¼ 0 for all J; proving (5.3). The ﬁnal statement in the lemma follows from (5.5). & Specializing the argument yielding (5.2) to N ¼ 1; we obtain the well-known identity Bl ð1 tÞ ¼ ð 1Þl Bl ðtÞ: Note that the latter, combined with (5.3), yields an alternative proof of (5.2). Combining Lemma 5.1 with our previous information on the polynomials PN;n and Pm;N;n ; we easily obtain the following formulas.

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Corollary 5.2. The above polynomials have the symmetries PN;n ðW ð1 xÞÞ ¼ ð 1ÞN PN;n ðW ðxÞÞ; Pm;N;n ðW ð1 xÞÞ ¼ ð 1Þm Pm;N;n ðW ðxÞÞ;

mAN;

ð5:6Þ ð5:7Þ

and admit the ‘‘Bernoulli representations’’ PN;n ðW ðxÞÞ ¼

X

gL;N;n

jLj¼N

Pm;N;n ðW ðxÞÞ ¼

X

BL ðxÞ ; L!

gL;m;N;n

jLj¼m

Here, the coefficients can be written Z gL;N;n ¼ @xL PN;n ðW ðxÞÞ dx;

gL;m;N;n ¼

IN

Proof. Using (3.9) and (4.5), we obtain Z @xJ PN;n ðW ðxÞÞ dx ¼ 0;

Z IN

BL ðxÞ : L!

@xL Pm;N;n ðW ðxÞÞ dx:

ð5:8Þ

ð5:9Þ

ð5:10Þ

0pjJjoN:

IN

Since by Proposition 3.2, PN;n ðW ðxÞÞ has degree at most N; Lemma 5.1 entails (5.8) and the integral formula (5.10) for gL;N;n : Likewise, (5.9) and the coefﬁcient formula (5.10) follow from (4.3). Finally, the symmetries (5.6) and (5.7) are plain from (5.2). & We recall that for m ¼ kn þ N with kAN; the polynomial Pm;N;n ðW ðxÞÞ equals zN;n ð k; W ðxÞÞ: In this case, representation (5.9) was obtained ﬁrst by Shintani [13, p. 398]. Moreover, Shintani not only obtained (5.8), but also stated an impressive formula for gL;N;n [16, p. 206]. It is not clear how he obtained the latter, and we have not tried to supply a proof. We sketch, however, a proof that if Npn; then PN;n ðwÞ is a sum of monomials of b

the kind wbi i wj j ; i.e., no products of three or more distinct wj ’s appear in PN;n ðwÞ: This fact is not clear from Shintani’s formula. For the proof we may assume N ¼ n; as the case Non follows from the difference equation. In this ‘‘equidimensional’’ case, the techniques in Sections 4 and 5 involving derivatives with respect to the xi can all be replaced by derivatives with respect to the wj : The integrals over I N are now replaced by integrals over the convex span P of the ai : Where we had W ðxÞ for x in I N ; we now have w in P: The advantage is that the w-partials @wJ w SðsÞ are very much simpler than (4.10). By studying these we ﬁnd that there are no terms of degree

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N involving three distinct wj ’s. Using the vanishing of integrals in degree less than N; one can show that there are no terms in PN;n ðwÞ of any degree involving three or more distinct wj ’s. The results in the following proposition are due to Shintani [16, pp. 204, 206, 210] when w ¼ W ðxÞ: Proposition 5.3. Setting A :¼ a1 þ a2 þ ? þ aN ; we have the identity GN;n ðwÞ ðGN;n ðA wÞÞð 1Þ

Nþ1

¼

n Y

GN ðwj ÞðGN ðAj wj ÞÞð 1Þ

Nþ1

;

ð5:11Þ

j¼1

where NX1; wACn ; and GN;n ðwÞ ¼ GN;n ðwja1 ; y; aN Þ;

GN ðwj Þ ¼ GN ðwj ja1j ; y; aNj Þ:

Moreover, for NX1; wACn and xACN ; we have zN;n ð0; wja1 ; y; aN Þ ¼ ð 1ÞN zN;n ð0; A wja1 ; y; aN Þ;

zN;n ð0; wja1 ; y; aN Þ ¼

zN ð0; W ðxÞja1 ; y; aN Þ ¼

ð5:12Þ

n 1X z ð0; wj ja1j ; y; aNj Þ; n j¼1 N

ð 1ÞN X aL BL ðxÞ; a1 a2 ?aN L L!

ð5:13Þ

N Y

aLi i :

ð5:14Þ

BJ :¼ BJ ð0Þ:

ð5:15Þ

aL :¼

i¼1

jLj¼N

Corollary 5.4. zN ð0; wja1 ; y; aN Þ ¼

N ð 1ÞN X wj X a J BJ ; a1 a2 ?aN j¼0 j! J J! jJj¼N j

Proof of Proposition 5.3. From (3.9) and (5.6) we obtain (5.11) for w ¼ W ðxÞ; with xACN : (Note A W ðxÞ ¼ W ð1 xÞ:) Now for NXn; the ai generically span Cn ; so we can generically write any wACn as w ¼ W ðxÞ for some xACN : By analyticity in the ai and w; (5.11) therefore holds whenever NXn: Then the case Non of (5.11) follows recursively from (1.15). Analogously, we obtain (5.12) from (5.7) with m ¼ N; the zeta recurrence (1.7) playing the role of the gamma recurrence (1.15). We now prove (5.13). For xACN ; let RN;n ðxÞ :¼ zN;n ð0; W ðxÞja1 ; y; aN Þ

n 1X z ð0; W ðxÞj ja1j ; y; aNj Þ; n j¼1 N

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where W ðxÞj ¼ W ðxja1 ; y; aN Þj ¼ W ðxja1j ; y; aNj Þ;

ð5:16Þ

cf. (1.6). In view of (5.9) and (5.10) with m ¼ N; we need only show that all of the integrals Z @xL RN;n ðxÞ dx; jLj ¼ N; ð5:17Þ IN

vanish. For this purpose, we turn to the proof of Proposition 4.1, whose notation we retain. Recall from (4.8) that bzN;n ðs; W ðxÞÞ ¼ zN;n ðs; W ðxÞÞ W ðxÞ SðsÞ ; xAI N : Since jLj ¼ NX1; this entails @ LbzN;n ð0; W ðxÞÞ ¼ @ L zN;n ð0; W ðxÞÞ; x

Thus we have Z IN

x

@xL zN;n ð0; W ðxÞÞ dx ¼

Z IN

xACN :

bL ð0Þ; @xLbzN;n ð0; W ðxÞÞ dx ¼ Q N;n

where we have now added the subscript N; n to the notation deﬁned in (4.7). bL ðsÞ when s ¼ 0; we need to examine the terms cL ðsÞ=ðnsÞ In order to compute Q N;n I in (4.12) as s-0: Recall from (4.10) that cLI ðsÞ is deﬁned by @xL W ðxÞ SðsÞ ¼

X

cLI ðsÞ

I jIj¼N

n Y

ðx1 a1j þ ? þ xN aNj Þ s Ij :

ð5:18Þ

j¼1

Since cLI ðsÞ=ðnsÞ vanishes at s ¼ 0 if cLI ðsÞ contains an s2 factor, it follows from (5.18) that cLI ðsÞ=ðnsÞ vanishes at s ¼ 0 unless N ¼ jIj ¼ Ij for some j: Applying this to N; n and to N; 1 we ﬁnd N X bL ð0Þ ¼ 1 bL ð0Þ: Q Q N;n n j¼1 N;1 Therefore, (5.13) holds when w ¼ W ðxÞ: In the same way as for the previous formulas (5.11) and (5.12), this entails (5.13) for all wADn and MADN;n : It remains to prove (5.14). To this end, we compute the coefﬁcient Z gL;N;N;1 ¼ @xL zN ð0; W ðxÞÞ dx: IN

Since n ¼ 1; we have aij ¼ ai1 ¼ ai and I ¼ I1 ¼ N; so (5.18) becomes @xL W ðxÞ s ¼ ð 1ÞN ðx1 a1 þ ? þ xN aN Þ s N

N Y i¼1

aLi i ðs þ i 1Þ:

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As a consequence, (4.12) yields Z IN

@xL zN ð0; W ðxÞÞ dx ¼ dN ð 1ÞN

N Y

aLi i ;

i¼1

where dN ¼ ðN 1Þ!

Z N 1 Cþ

ðs1 a1 þ ? þ sN aN Þ N ds:

Note that dN depends on the ai and N; but not on L: We have thus far proved zN ð0; W ðxÞja1 ; y; aN Þ ¼ dN ð 1Þ

X Q N a Li i¼1 i BL ðxÞ: L! jLj¼N

N

ð5:19Þ

Rather than compute dN directly from the integral above, we shall use the difference equation (1.7) to relate dN to dN 1 : Note that d1 ¼ 1=a1 ; since Cþ0 ¼ f1g: In (5.19) we let x0 :¼ ðx1 ; y; xN 1 Þ be arbitrary, but take ﬁrst xN ¼ 1; then xN ¼ 0; and subtract to get zN ð0; W ððx0 ; 1ÞÞja1 ; y; aN Þ zN ð0; W ððx0 ; 0ÞÞja1 ; y; aN Þ ¼ zN ð0; W ððx0 ; 0ÞÞ þ aN Þ zN ð0; W ððx0 ; 0ÞÞÞ ¼ zN 1 ð0; W ðx0 Þja1 ; y; aN 1 Þ:

ð5:20Þ

Write an N-multi-index L of weight N as ðL0 ; iL0 Þ; with L0 an ðN 1Þ-multi-index and iL0 ¼ N jL0 j: Then, from (5.19) and (5.20), 0

zN 1 ð0; W ðx ÞÞ ¼ dN ð 1Þ

N

X jL0 jpN

QN 1 l¼1

L0

al l

L0 !

i

BL0 ðx0 Þ

aNL0 ðBi 0 ð1Þ BiL0 ð0ÞÞ: i L0 ! L

At face value, the latter substitution seems to complicate matters. The point is, however, that Bl ð1Þ Bl ð0Þ ¼ 0 unless l ¼ 1; in which case B1 ð1Þ B1 ð0Þ ¼ 1: Before proving this assertion, we show that it entails (5.14). Indeed, it yields 0

N

zN 1 ð0; W ðx Þja1 ; y; aN 1 Þ ¼ aN dN ð 1Þ

X jL0 j¼N 1

QN 1 l¼1

L0 !

L0

al l

BL0 ðx0 Þ:

Comparing this with (5.19) with N replaced by N 1; we obtain aN dN ¼ dN 1 : It follows that in (5.19) we have dN ¼ ða1 a2 ?aN Þ 1 ; so that (5.14) results.

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Finally, to prove the assertion, we use (1.29) to obtain N X ul ½ðBl ð1Þ Bl ð0Þ ¼ u; l! l¼0

whence its validity is clear.

&

Proof of Corollary 5.4. In (5.14) let xi ¼ 0 for ioN and xN ¼ w=aN ; so that W ðxÞ reduces to w: Then we ﬁnd zN ð0; wÞ ¼

N ð 1ÞN X alN kl Bl ðw=aN Þ; a1 a2 ?aN l¼0 l!

X QN 1 aLi i i¼1 BL ; L! L

kl :¼

ð5:21Þ

jLj¼N l

where L runs over ðN 1Þ-multi-indices. Using l X l! Bl j t j ; Bl ðtÞ ¼ ðl jÞ!j! j¼0 formula (5.15) follows from (5.21) on reversing the order of sums over l and j: (Note that the well-known expression above for Bl ðtÞ is readily proved: the rhs has the differentiation property (1.32) and coincides with the lhs at t ¼ 0:) & Possibly, Barnes was aware of explicit formula (5.15). However, he only wrote zN ð0; wÞ as a multiple Bernoulli polynomial, cf. Eq. (3.10) in [11]. From (5.15) we obtain after some calculation z1 ð0; wja1 Þ ¼

z2 ð0; wja1 ; a2 Þ ¼

z3 ð0; wja1 ; a2 ; a3 Þ ¼

1 w ; 2 a1

1 ð6w2 6ða1 þ a2 Þw þ a21 þ a22 þ 3a1 a2 Þ; 12a1 a2

ð5:22Þ

ð5:23Þ

1 ð 4w3 þ 6ða1 þ a2 þ a3 Þw2 24a1 a2 a3 ð2a21 þ 2a22 þ 2a23 þ 6a1 a2 þ 6a1 a3 þ 6a2 a3 Þw þ a21 a2 þ a21 a3 þ a22 a1 þ a22 a3 þ a23 a1 þ a23 a2 þ 3a1 a2 a3 Þ:

ð5:24Þ

6. Applications of the Raabe formula to certain inﬁnite products A glance at the Dirichlet series (1.5) deﬁning zN;n shows that restricting the variables wj and akj to the right half-plane is somewhat artiﬁcial. Indeed, for

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ReðsÞ4N=n the series also converges whenever ReðeiWj wj Þ; ReðeiWj akj Þ40; with Wj any angle in ð p; p and j ¼ 1; y; n: (Of course, this involves a ﬁxed choice of the P s logarithm branch used to deﬁne the complex powers ðwj þ N k¼1 mk akj Þ :) For n ¼ 1; this was the setting chosen by Barnes [3] to deﬁne his multiple gamma function. As will be clear from the following, this more general situation can be handled by analytic continuation in the vectors w and a1 ; y; aN : In order to steer clear of multivaluedness, however, we restrict attention to vectors W in ð p=2; p=2Þn from now on. First, we recall from Proposition 2.1 and (1.3) that the function kðs; w; a1 ; y; aN Þ :¼ zN;n ðs; wj a1 ; y; aN Þ=pN;n ðsÞ

ð6:1Þ

is holomorphic in C T; where T is the tube-like domain T :¼ fðw; a1 ; y; aN ÞADNþ1 g; n

Dn :¼ fvACn j Reðvj Þ40g:

For vACn we now introduce vðWÞ :¼ ðeiW1 v1 ; y; eiWn vn Þ: Next, we deﬁne domains Dn ðWÞ :¼ fvACn j vðWÞADn g; TðWÞ :¼ fðw; a1 ; y; aN ÞADn ðWÞNþ1 g; Text :¼

[

TðWÞ:

ð6:2Þ

WAð p=2;p=2Þn

The W-restriction ensures that none of the wj ; akj in the ‘‘extended tube’’ Text belongs to ð N; 0 : Therefore, Text is a simply connected domain, on which no multivaluedness can occur. We assert that k in (6.1) has a holomorphic extension to C Text : To show this, we ﬁx WAð p=2; p=2Þn and deﬁne a function kW on the domain C TðWÞ by Pn is W j¼1 j kðs; wðWÞ; a ðWÞ; y; a ðWÞÞ: kW ðs; w; a1 ; y; aN Þ :¼ e ð6:3Þ 1 N Obviously, kW is holomorphic in C TðWÞ: On the subdomain fReðsÞ4N=ng ðT-TðWÞÞ; kW coincides with k; as is clear from the series representation (1.5). Thus our assertion readily follows. Multiplying k by pN;n ðsÞ; we obtain analyticity properties of zN;n in C Text that are plain. They entail in particular that all of our previous results regarding zN;n have

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generalizations to the extended tube Text (6.2). Since s ¼ 0 is a regular value, the gamma functions are well deﬁned and holomorphic on Text as well, and Proposition 2.2 has an immediate generalization that need not be spelled out. Likewise, our previous results on GN;n can be analytically continued to all wACn and a1 ; y; aN in Dn ðWÞ for any WAð p=2; p=2Þn ; in particular, this is the case for the generalized Raabe formula (1.25), whose extended version we will have occasion to invoke shortly. Note that the analog of (6.3) reads izN;n ð0;wja1 ;y;aN Þ

GN;n;W ðwja1 ; y; aN Þ ¼ e

n P j¼1

Wj

GN;n ðwðWÞja1 ðWÞ; y; aN ðWÞÞ:

(The s ¼ 0 value of zN;n does not depend on W; cf. (6.3).) We can now relate Barnes’ multiple gamma function to certain inﬁnite products. Proposition 6.1. Let a1 ; y; aN be N complex numbers in the upper half-plane, and set a :¼ ða1 ; y; aN Þ;

ma :¼

N X

m k ak ;

m ¼ ðm1 ; y; mN ÞANN :

k¼1

Then we have the following equality between meromorphic functions of w: GNþ1 ðwj1; aÞGNþ1 ð1 wj1; aÞ ¼ e pizNþ1 ð0;wj1;aÞ

Y

ð1 e2piðwþmaÞ Þ 1 :

ð6:4Þ

N

mAN

When N ¼ 0; the above formula amounts to 1

e pið2 wÞ G1 ðwj1ÞG1 ð1 wj1Þ ¼ 1 e2piw

or

GðwÞGð1 wÞ ¼

p ; sinðpwÞ

as can be seen from Lerch’s formula (1.23). The case N ¼ 1 of Proposition 6.1 was proved by Barnes [2, p. 376], and re-proved by Shintani [18, p. 196]. For N41 we have not been able to ﬁnd (6.4) in the literature. Since the ratio of the two sides is easily seen to have neither zeroes nor poles, the main point of Proposition 6.1 is the exact determination of the entire function appearing in the exponential. From (5.12), (5.23) and (6.4) with N ¼ 1; one obtains a relation between Barnes’ double gamma function and the modular functions appearing in Kronecker’s second limit formula. Namely [18, Proposition 2], G2 ðwj1; tÞG2 ð1 wj1; tÞG2 ð1 þ t wj1; tÞG2 ðw tj1; tÞ ¼ epiTðwÞ

N Y j¼0

ð1 e2piw e2pijt Þ 1 ð1 e 2piw e2piðjþ1Þt Þ 1 ;

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where t is in the upper half-plane, wAC; and the quadratic polynomial T is given by 1 t w2 w þ 16 : TðwÞ ¼ 2z2 ð0; wj1; tÞ ¼ w 2 6 t Proof of Proposition 6.1. The analyticity properties established above entail that we need only prove (6.4) for w in the half-strip ReðwÞAð0; 1Þ; ImðwÞA½0; NÞ; and a1 ; y; aN Aið0; NÞ: Assuming this from now on, we begin by observing that the second logarithmic derivative of the well-known N ¼ 0 case of (6.4), combined with the relation of G1 ðwj 1Þ to the Hurwitz zeta function Hðs; wÞ; yields the identity X

1

kAZ

ðw þ kÞ2

¼

p2 ; sin2 ðpwÞ

ReðwÞAð0; 1Þ:

(Of course, this is another well-known identity.) Now we replace w above by w þ ma and take the Nth w-derivative. Summing over mANN ; we obtain an identity of convergent series, namely, ð 1ÞN ðN þ 1Þ!

X X

1

mANN kAZ

ðw þ k þ maÞNþ2

¼

dN X p2 : dwN sin2 pðw þ maÞ mANN

ð6:5Þ

Taking the s-derivative at s ¼ 0 of (3.1) with J ¼ N þ 2 and n ¼ 1; we recognize the lhs of (6.5) as d Nþ2 logðGNþ1 ðwj1; aÞGNþ1 ð1 wj1; aÞÞ: dwNþ2 The rhs is minus the ðN þ 2Þth logarithmic derivative of the inﬁnite product Y fNþ1 ðwÞ :¼ ð1 e2piðwþmaÞ Þ: ð6:6Þ mANN

Thus, (6.4) is correct up to a factor epðwÞ ; where p is a polynomial of degree at most N þ 1: We can now prove (6.4) using the Raabe-type formula (1.25) and induction on N: To this end we introduce piQNþ1 ðwj aÞ :¼ log GNþ1 ðwj1; aÞ þ log GNþ1 ð1 wj1; aÞ þ log fNþ1 ðwÞ:

ð6:7Þ

Here, the logarithms of the gamma functions are the ones ﬁxed by the s-derivative of zN at s ¼ 0; whereas the logarithm branch for fNþ1 ðwÞ is ﬁxed by requiring that when we let w converge to iN in the above half-strip, the limit vanishes. From the foregoing discussion, we already know that QNþ1 ðwÞ is a polynomial. It remains to prove that QNþ1 ðwÞ ¼ zNþ1 ð0; wÞ: As discussed above, this is true for N ¼ 0: A short calculation, using (6.6), (6.7), (1.15) and the inductive hypothesis

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QN ¼ zN ; shows QNþ1 ðw þ aN Þ ¼ QNþ1 ðwÞ QN ðwj a0 Þ ¼ QNþ1 ðwÞ zN ð0; wj 1; a0 Þ; where a0 ¼ fa1 ; y; aN 1 g and NX1: Since zNþ1 ð0; wj1; aÞ is another polynomial satisfying the same ordinary difference equation as QNþ1 ; we have QNþ1 ðwÞ ¼ zNþ1 ð0; wj1; aÞ þ c for some constant c independent of w: For x ¼ ðx0 ; x1 ; y; xN Þ; we substitute in (6.7) w ¼ W ðxÞ ¼ W ðxj1; aÞ ¼ x0 þ x1 a1 þ x2 a2 þ ? þ xN aN ;

xAI Nþ1 :

From (4.1) we ﬁnd c¼

Z

QNþ1 ðW ðxÞj aÞ dx:

I Nþ1

The vanishing of c now follows from (6.7) and the vanishing of the following three integrals: Z log GNþ1 ðW ðxj1; aÞj1; aÞ dx; K :¼ I Nþ1 Z L :¼ log GNþ1 ðð1 W ðxj1; aÞÞj1; aÞ dx; I Nþ1 Z M :¼ log fNþ1 ðW ðxj1; aÞÞ dx: ð6:8Þ I Nþ1

The vanishing of K is a direct application of the Raabe formula (1.25). Switching now to L; note that 1 W ðxj1; aÞ ¼ 1 x0 x1 a1 x2 a2 ? xN aN ¼ W ðyj1; aÞ; where y :¼ ðy0 ; y1 ; y; yN Þ; y0 :¼ 1 x0 ; yk :¼ xk ; 1pkpN: Therefore, changing variables from x to y in deﬁnition (6.8) of L; we obtain Z Z log GNþ1 ð1 W ðxj1; aÞj1; aÞ dx ¼ log GNþ1 ðW ðyj1; aÞj1; aÞ dy; I Nþ1

I Nþ1

which again vanishes by the Raabe formula. The vanishing of M can be seen as follows. Writing xAI Nþ1 as x ¼ ðt; yÞ with 0oto1 and yAI N ; we have M¼

Z

Z IN

1

t¼0

log fNþ1 ðt þ W ðyj aÞÞ dt dy:

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Now the integrand has no singularities in the half-strip ReðtÞAð0; 1Þ; ImðtÞA½0; NÞ; and it has period 1 in t: Thus we may shift the contour of the t-integral to iN; and deduce it vanishes. & From Proposition 6.1 we can derive a formula for the elliptic gamma function given by Gðr; a; b; wÞ :¼

N Y 1 e 2irw e rað2jþ1Þ e rbð2kþ1Þ ; 1 e2irw e rað2jþ1Þ e rbð2kþ1Þ j;k¼0

ð6:9Þ

where r40 and ReðaÞ; ReðbÞ40 [10, p. 1104]. Corollary 6.2. Setting a :¼ ia; b :¼ ib and g :¼ ða þ bÞ=2; we have Gðp; a; b; wÞ ¼ epiRðwÞ

G3 ðw þ gj 1; a; bÞG3 ð1 w gj 1; a; bÞ ; G3 ð w þ gj 1; a; bÞG3 ð1 þ w gj 1; a; bÞ

where RðwÞ is the cubic polynomial RðwÞ ¼ z3 ð0; w þ gj1; a; bÞ z3 ð0; w þ gj 1; a; bÞ ¼

w3 a2 þ b2 þ 2 w: þ 12ab 3ab

Proof. This can be veriﬁed from Proposition 6.1, deﬁnition (6.9) and Eq. (5.24) by a simple calculation. & Note that G is invariant under r; a; b; w-rt; a=t; b=t; w=t; so ﬁxing r ¼ p in Corollary 6.2 is no restriction.

Acknowledgments It is a pleasure to record our thanks to Jan Felipe van Diejen for his many helpful comments.

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