Narukawa a - The Modular Properties and the Integral Representations of the Multiple Elliptic Gamma Functions - Avd. in Math. 189 (2004), 247-267

8/9/2019 Narukawa a - The Modular Properties and the Integral Representations of the Multiple Elliptic Gamma Functions - A

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

Advances in Mathematics 189 (2004) 247267

The modular properties and the integral

representations of the multiple elliptic

gamma functions

Atsushi Narukawa1

Department of Mathematics, School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan

Received 6 June 2003; accepted 26 November 2003

Communicated by P. Etingof

Abstract

We show the modular properties of the multiple elliptic gamma functions, which are an

extension of those of the theta function and the elliptic gamma function. The modularproperty of the theta function is known as Jacobis transformation, and that of the elliptic

gamma function was provided by Felder and Varchenko. In this paper, we deal with the

multiple sine functions, since the modular properties of the multiple elliptic gamma functions

result from the equivalence between two ways to represent the multiple sine functions as

infinite products.

We also derive integral representations of the multiple sine functions and the multiple

elliptic gamma functions. We introduce correspondences between the multiple elliptic gamma

functions and the multiple sine functions.

r 2003 Elsevier Inc. All rights reserved.

MSC: 33B15; 33D05; 33E30; 11F03

Keywords: Modular property; Multiple elliptic gamma function; Multiple sine function; q-Shifted

factorial; Generalized q-polylogarithm

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E-mail address: [email protected] author recently transferred to The Dai-ichi Mutual Life Insurance Company, Tokyo 100-8411,

Japan.

0001-8708/$ - see front matterr 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2003.11.009


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1. Introduction

The theta function y0z; t and the elliptic gamma function Gz; t; s are defined by

infinite products

y0z; t YNj0

1 e2pi j1tz1 e2pi jtz;

Gz; t; s YN

j;k0

1 e2pi j1tk1sz

1 e2pi jtksz;

then Gz; t; s satisfies the difference equation Gz t; t; s y0z; sGz; t; s:The elliptic gamma function was originally constructed by Ruijsenaars [10] as a

unique solution of difference equations which include the theta function as above.After his work, Felder and Varchenko [3] derived the modular property of this

function,

Gz

s;t

s;

1

s

epiQz;t;sG

z s

t;

s

t;

1

t

Gz; t; s; 1

where

Qz; t; s z3

3ts

t s 1

2ts

z2 t2 s2 1 3ts 3t 3s

6ts

z

t s 1ts t s

12ts:

This formula is an extension of Jacobis transformation, in which the group

SL2;ZrZ2 acts on the parameter of the theta function. Felder and Varchenkodeduced (1) from the modular properties ofy0z; t and the special value ofGz; t; s:

They also gave a cohomological interpretation to this formula with SL3;ZrZ3:On the other hand, Nishizawa [9] constructed a hierarchy of meromorphic

functions which includes the theta function and the elliptic gamma function. He

called these new functions the multiple elliptic gamma functions Grz j t0; y; tr;which are considered as an elliptic analogue of the multiple gamma functions. They

are defined by certain infinite products called q-shifted factorials x;%

qrN

: They

satisfy functional relations, such as

Grz tj j t0; y; tr Gr1z j t0; y; $tj; y; trGrz j t0; y; tr:

Conversely he characterized these functions with above relations and initial values.

Our main purpose of this paper is to derive the modular properties of the multiple

elliptic gamma functions Grz j t0; y; tr while we discuss the properties of the

multiple sine functions Srz j o1; y;or and the multiple Bernoulli polynomialsBr;nz j o1; y;or: The hierarchy of the multiple sine functions is defined by Barnesmultiple gamma functions. Those have been studied by Shintani [12], Kurokawa

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[7,8]. The multiple Bernoulli polynomials are attached to the multiple zeta functions

and the multiple gamma functions as in [1,11].

We introduce integral representations and infinite product representations of the

multiple sine functions. Then it is shown that there are two ways to represent them asan infinite product. From the equivalence between them, the modular properties of

Grz j t0; y; tr are obtained. For example, (1) is derived from the two representa-tions of S3z j o1;o2;o3; namely

S3z j o1;o2;o3

exp pi

6B33z j o1;o2;o3

& '

YN

j;k0

1 e2piz=o1 j1o2=o1k1o3=o1

1 e2piz=o3jo1=o3ko2=o3

1 e2piz=o2jo1=o2k1o3=o2

exp pi

6B33z j o1;o2;o3

& '

YN

j;k0

1 e2piz=o1jo2=o1ko3=o11 e2piz=o3 j1o1=o3k1o2=o3

1 e2piz=o2 j1o1=o2ko3=o2:

Substituting o1 t; o2 s; o3 1; we have (1) and the fact

Qz; t; s 13 B33z j t; s; 1:

In general, we use q-shifted factorials x;%

qrN

to describe Grz j t0; y; tr and

Srz j o1; y;or: The q-shifted factorial is the exponential of the generalized q-polylogarithm. The general result which we prove is the following theorem.

Theorem. If rX2; Imojoka0; then the multiple elliptic gamma function satisfies the

identity

Yr

k1

Gr2zok

o1ok; y;$ok

ok; y; o

r

ok

exp 2pi

r!Brrz j

%o

& ':

The remaining part of this paper is devoted to investigate the integral

representations of Grz j t0; y; tr: We recall the results in [3,9] again, and regardGrz j t0; y; tr as infinite products of Sr1z j o1; y;or1:

The paper is organized as follows: In Sections 2 and 3, we review the definition and

the properties of Nishizawas multiple elliptic gamma functions and those of the

multiple Bernoulli polynomials. Then in Section 4, we introduce the integral

representations and the infinite product representations of the multiple sinefunctions. In Section 5, we prove the modular properties of the multiple elliptic

gamma functions. In Section 6, we introduce the integral representations of the

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multiple elliptic gamma functions, and these integrals show that the multiple elliptic

gamma functions are described as infinite products of the multiple sine functions.

2. The multiple elliptic gamma functions Grz j%s

In this section, we review the multiple elliptic gamma functions Grz j%t according

to Nishizawa [9].

Let x e2piz; qj e2pitj for zAC and tjAC R 0pjpr; and

%

q q0; y y; qr;

%q

j q0; y; qj; y; qr;

%

q j q0; y; q1

j ; y; qr;

%

q1 q10 ; y y; q1r ;

where qj means the excluding of qj: When Im tj40 for all j; define the q-shifted

factorial

x; %q

r

N YN

j0;y; jr0 1 xq

j0

0?q

jr

r :

This infinite product converges absolutely when jqjjo1: Thus this function is aholomorphic function with regard to z; whose zeros are

z t0Zp0 ? trZp0 Z:

In general, we can define the q-shifted factorial for tjAC R as follows: WhenIm t0; y; Im tk1o0 and Im tk; y; Im tr40; that is, jq0j; y; jqk1j41 and

jqkj; y; jqrjo1; we define

x;%

qrN

fq10 ?q1k1x; q

10 ; y; q

1k1; qk; y; qr

rN

g1k

2

YN

j0;y; jr0

1 xqj010 ?qjk11k1 q

jkk?q

jrr

( )1k: 3

More general definition can be done in a similar way as the values are not changedunder the permutation of q0; y; qr: In this definition, x;

%

qrN

is a meromorphic

function of z satisfying the following functional equations.

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Proposition 1.

x; %q

r

N

1

q1j x;%

q jrN

; qjx; %q

r

N

x;

%

qrN

x;%

q jr1N

:

We next denote

%t t0; y y; tr;

%t j t0; y; $tj; y; tr;

%t j t0; y; tj; y; tr;

%t t0; y y; tr;

j%tj t0 ? tr

and define the multiple elliptic gamma function

Grz j%t x1q0?qr;

%

qrN

fx;%

qrN

g1r

4

fx1;%

q1rN

g1r1

fx;%

qrN

g1r

: 5

Grz j%t is defined for tjAC R from the general definition ofx;

%

qrN: The hierarchy

of Grz j%t includes the theta function y0z; t and the elliptic gamma function

Gz; t; s which appeared in [10,3]. When Im t; Im s40; recall the definition

y0z; t YNj0

1 e2pi j1tz1 e2pi jtz G0z j t;

Gz; t; s YN

j;k0

1 e2pi j1tk1sz

1 e2pi jtksz G1z j t; s:

The definition of x;%

qrN

and Proposition 1 imply the functional equations:

Grz 1 j%t Grz j

%t; 6

Grz tj j%t Gr1z j

%t jGrz j

%t; 7

Grz j%t

1

Grz tj j%t j

; 8

Grz j %t

1

Grz j%t

; 9

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Grz j%tGrz j

%t j

1

Gr1z j%t j

: 10

Grz j %t can be expressed as an infinite product directly by (8) and (3) for anytjAC R:The zeros and poles are easily observed if Im tj40 for all j: When r is even,

Grz j%t is holomorphic on C; and zeros are written as follows:

zeros z t0Zp0 ? trZp0 Z;

z t0ZX1 ? trZX1 Z:

When r is odd, Grz j%t is meromorphic on C with poles and zeros written as follows:

poles z t0Zp0 ? trZp0 Z;

zeros z t0ZX1 ? trZX1 Z:

In particular, Grz j%t has no poles and no zeros in the domain f0oIm zoIm j

%tjg if

Im tj40 for all j:

When r 0; G0z j t0 means the theta function y0z; t: As we know, y0z; t

possesses the periodicity y0z 1; t y0z; t; y0z t; t e2piz1=2y0z; t and

the modular property

y0z

t;

1

t

exp pi

z2

t

z

t z

t

6

1

6t

1

2

& 'y0z; t: 11

The modular property ofr 1 case has been already described in (1) referring to [3].We derive the modular properties for general Grz j

%t in this paper.

3. The multiple Bernoulli polynomials Br;nz j%x

For zAC;%o o1; y;or; ojAC f0g; we define the multiple Bernoulli

polynomials Br;nz j%o with a generating function

treztQrj1 e

ojt 1XNn0

Br;nz j%o

tn

n!:

They essentially appeared in [1] and play an important role to study the multiple zeta

functions and the multiple gamma functions as in the next section.

Br;nz j%o is a polynomial of degree n in z and is symmetric in o1; y;or: It is easy

to show that

Br;ncz j c%o cnrBr;nz j

%o 8cAC f0g; 12

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Br;nj%oj zj

%o 1nBr;nzj

%o; 13

Br;nz oj j %o Br;nz j %o nBr1;n1z j %o

j; 14

Br;nz j%o j Br;nz oj j

%o; 15

Br;nz j%o Br;nz j

%oj nBr1;n1z j

%o j; 16

d

dzBr;nz j

%o nBr;n1z j

%o 17

where

c%o co1; y; cor;

j%oj o1 ? or;

%o j o1; y; $oj; y;or;

%oj o1; y; oj; y;or

and $oj means the excluding ofoj: In particular, Br;nz Br;nz j 1; y; 1 obeys

rBr1;nz 1 r nBr;nz nzBr;n1z;

Br1;rz z 1?z r r!z 1

r

:

For example, we can see that

B11z j o1 z

o1

1

2;

B22z j o1;o2 z2

o1o2

o1 o2o1o2

z o21 o

22 3o1o2

6o1o2;

B33z j o1;o2;o3 z3

o1o2o3

3o1 o2 o3

2o1o2o3z2

o21 o

22 o

23 3o1o2 3o2o3 3o3o1

2o1o2o3

z

o1 o2 o3o1o2 o2o3 o3o1

4o1o2o3:

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4. Definition and properties of the multiple sine functions Srz j%x

4.1. Definition of the multiple sine functions Srz j

%

o

Now suppose that the points representing o1; y;orAC all lie on the sameside of some straight line through the origin. (Usually we suppose o1; y;orACall lie on the right half plane.) In this case, the multiple zeta function is defined

by the series

zrs; z j%o

XNn1;y;nr0

1

n1o1 ? nror zs

for zAC; Re s4r; where the exponential is rendered one-valued. This series isholomorphic in the domain fRe s4rg; and it is analytically continued to sAC: Sinceit is holomorphic at s 0; we can next define the multiple gamma function by

Grz j%o exp

@

@szrs; z j

%o

s0

: 18

Now we define the multiple sine function by the form

Srz j%o Grz j

%o1Grj

%oj z j

%o1

r

: 19

The above definition of zrs; z j%o is due to Barnes [1], and the definitions of

Grz j%o and Srz j

%o are due to Kurokawa [8] and JimboMiwa [4]. Barnes

Grz j%o is slightly different in the coefficient called Barnes modular constant.

In particular, the double sine function S2z j o1;o2 have been studied to constructsolutions or operators of certain equations of mathematical physics as in [4] or [5].

4.2. Integral representations of Srz j%o

Proposition 2. (i) Srcz j c%o Srz j

%o for all cAC f0g:

(ii) When 0oRe oj 8j and 0oRe zoRe j%oj; Srz j

%o has the integral representa-

tions

Srz j%o exp 1r

pi

r!Brrz j

%o 1r

ZRi0

ezt

tQr

j1eojt 1

dt

( )20

exp 1r1pi

r!Brrz j

%o 1r

ZRi0

ezt

tQr

j1 eojt 1

dt

( ); 21

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where the contours are taken as following figures:

(iii) When 0oRe oj 8j; Srz j%o has no poles and no zeros in the domain

f0oRe zoRe j%ojg:

Proof. Let L be the half-line from the origin, let %L be the half-line conjugate to L

with respect to the real axis, and let %L> be the line at right angles to %L: Further we

take a contour L which embraces L; and other contours C0; C as following figures.

Now we assume all of z;o1; y;or and %L lie on the same side with respect to %L>:

Then as in [1], Grz j%o has an integral representation

Grz j%o exp

1

2pi

ZL

eztflog t gg

tQr

j1 1 eojt

dt

( );

where log t is rendered one-valued by the cross-cut along L; and log t is realwhen t is in Ro0 (real and negative). g is Eulers constant.

Under this assumption, study Srcz j c%o for any cAC with ceL: Then all of cz;

co1; y; cor lie on the same side with a half-line c %L fct j tA %Lg: Let c1L be the

half-line conjugate to c %L; and let c1 L be a contour which embraces the half-line

c1L: Then we obtain the rotated expression

Grcz j c%o exp

1

2pi

Zc1 L

ecztflog t gg

tQr

j1 1 ecojt

dt

( );

where the cross-cut of log t is the half-line c1L; and log t is real when tARo0:

Further changing t into c1t and changing the branch of the logarithm, we get

Grcz j c

%

o exp1

2piZ

L

eztflog t log c gg

tQr

j1 1 eojtdt( );

where the cross-cut of log t is the half-line L; and log t is real when tARo0:

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If we denote jt ezt

tQr

j1eojt1

; the above expression is rewritten into

Grcz j c%o exp

ZL1r1jtflog t log c gg

2pidt

& ':

Moreover assuming that j%oj z lies on the same side with %L as well as z and

%o; we

get

Grcj%oj cz j c

%o exp

ZL

jtflogt logc gg

2pidt

& ':

Thus by definition (19),

Srcz j c%o exp 1r

ZL

fjt jtgflogt log c gg

2pidt

& ':

When we consider the integralR

Ljt jt dt; L can be replaced by C0 because

jt and jt are rapidly decreasing along the half-line L: We note that jt jt is an even function, then we haveZ

L

jt jt dt

ZC0

jt jt dt 0:

They yield

Srcz j c%o exp 1r

ZL

fjt jtglog t

2pidt

& ': 22

The right-hand side of this formula is independent of c: This means that Srcz j c%o

coincides with Srz j%o under the assumption that z and j

%oj z lie on the same side

with %L: Nevertheless the analytic continuation certifies that Srcz j c%o Srz j

%o for

any z except the poles. Another assumption ceL is not essential since we couldchoose a line L such that ceL for any c: We conclude (i).

Next we have to show (20) of (ii). Suppose 0oRe oj; 0oRe zoRe j%oj; c 1;

L RX0; L C: We shall transform the integral of (22) after fixing the branch ofthe logarithm such that the cross-cut of log t is the half-line RX0; and log t isreal when tARo0:

Set the contours as above figure, then stretching the contour C impliesZC

jtlogt

2pidt

ZRie

jtlogt

2pidt

ZRie

jtlogt

2pidt:

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By changing t into t without changing the branch of the logarithm, we have

ZRie

jtlogt

2pidt Z

Rie

jtlogt pi

2pidt;Z

Rie

jtlogt

2pidt

ZRie

jtlogt pi

2pidt:

ThusZC

jtlogt

2pidt

ZRie

jtlogt pi

2pidt

ZRie

jtlogt pi

2pidt:

On the other hand, stretching the contour C impliesZC

jtlog t

2pidt

ZRie

jtlog t

2pidt

ZRie

jtlog t

2pidt:

Consequently we obtainZC

fjt jtglogt

2pidt

1

2

ZRie

jt dt 1

2

ZRie

jt dt

1

2 ZRi0

jt dt 1

2 ZRi0

jt dt ZC0

jt dt& '

ZRi0

jt dt pi

r!Brrz j

%o;

since the residue ofjt at t 0 is 1r!

Brrz j%o: (20) is shown.

Eq. (21) follows easily from (20). (iii) is due to (ii) because the integrals of (ii) are

bounded. &

The following well-known formulae hold from the integral representations of

Srz j%o:

Srz oj j%o Sr1z j

%o j1Srz j

%o; Srz j

%oSrj

%oj z j

%o1

r

1:

4.3. Definition of the generalized q-polylogarithm Lir2x;%

q

To describe the infinite product representations of Srz j%o; we start with a

definition and a lemma. Now let x e2piz; qj e2pitj: When Im z40 and

Im tja0 0pjpr; Nishizawa [9] defined the generalized q-polylogarithm

Lir2x;%

q XNn1

xn

nQr

j0 1 qn

j:

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This series converges absolutely and then it is holomorphic in z: This function is ageneralization of Kirillovs quantum polylogarithm [6], whose parameters q0; y; qrare all equal. If r 0; the above series is the quantum dilogarithm in [2].

We can see the functional equation

Lir2x;%

q Lir2q1

j x;%

q j: 23

Let us review the following fundamental lemma verified in [9].

Lemma 3. Let Im z40; Im tja0; that is, jxjo1; jqjja1; then

x;%

qrN

expLir2x;%

q:

Proof. First we discuss the case when jqjjo1 8j: Applying the formula 1 x

expP

N

n1xn

n; it follows that:

expLir2x;%

q exp XNn1

xn

nQr

j0 1 qn

j

( )

exp XNn1

XNj0;y; jr0

xnqnj00 ?q

njrr

n

( )

YN

j0;y; jr0

exp XNn1

xq

j0

0?q

jr

r

n

n

( )

YN

j0;y; jr0

1 xqj00?qjrr

x;%

qrN

:

Next it is enough to discuss the case when jq0j; y; jqk1j41; and jqkj; y; jqrjo1:The first case and (2), (23) implies

x;%

qrN

fq10 ?q1k1x; q

10 ; y; q

1k1; qk; y; qr

rN

g1k

fexpLir2q10 ?q

1k1x; q

10 ; y; q

1k1; qk; y; qrg

1k

fexp1k Lir2x; q0; y; qk1; qk; y; qrg1k

expLir2x;%

q: &

4.4. Infinite product representations of Srz j%o

We need the following lemma.

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Lemma 4. If 0oRe zoRe j%oj; Im z40 and Im z4Im j

%oj; there exists a real series

fang such that

limn-N

an N and limn-N

ZRian

ezt

tQr

j1 eojt 1

dt 0:

Similarly if0oRe zoRe j%oj; Im zo0 andIm zoIm j

%oj; there exists a real series fang

such that

limn-N

an N and limn-N

ZRian

ezt

tQr

j1 eojt 1

dt 0;

where the contour is drawn in the following figure.

There exist a small E40 and a real series fang such that the distances from eachpoles to any contours R ian are more than E: We reach the above lemma by

estimating the absolute value of the integrand.Now we set xk e

2piz=ok; qjk e2pioj=ok; qk q1k; y; qkk; y; qrk and q

1k

q11k; y; q1kk; y; q

1rk ; then we are ready to prove the formula on Srz j

%o:

Proposition 5. If rX2; Imojoka0; then Srz j

%o has the following infinite product

representations:

Srz j%o exp 1r

pi

r!Brrz j

%o

& 'Y

r

k1

xk; qkr2N

24

exp 1r1pi

r!Brrz j

%o

& 'Yrk1

x1k ; q1k

r2N

: 25

Proof. From Proposition 2(i) and (12), it is enough to discuss the case 0oRe oj for

all j: We need to evaluate the integrals of Proposition 2(ii) by the residue formula.We first compute that

Rest

2pinok

ezt

tQr

j1 eojt 1

xnk

2pinQr

j1;jak qn

jk 1

:

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For the convergency of the series, we restrict the domain of z to

zAC j 0oRe zoRe j%oj; 0oIm z; Im j

%ojoIm z; Im

z

ok

40 8k& ';which is not void when Re oj40: Then we can use Lemma 4, and add up theresidues in fIm t40g; namely

exp 1rZRi0

ezt

tQr

j1 eojt 1

dt

( )

exp 1r2piXrk1

XNn1

Rest

2pinok

ezt

t

Qr

j1 eojt 1

8

Documents

Narukawa a - The Modular Properties and the Integral Representations of the Multiple Elliptic Gamma Functions - Avd. in Math. 189 (2004), 247-267