An elliptic analogue of the multiple gamma function

Nishizawa, Michitomo

doi:10.1088/0305-4470/34/36/320

Cited by 23 publications

(31 citation statements)

References 14 publications

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“…The remaining part of this paper is devoted to investigate the integral representations of G r ðz j t 0 ; y; t r Þ: We recall the results in [3,9] again, and regard G r ðz j t 0 ; y; t r Þ as infinite products of S rþ1 ðz j o 1 ; y; o rþ1 Þ:…”

Section: Introductionmentioning

confidence: 99%

“…In this section, we review the multiple elliptic gamma functions G r ðz j % tÞ according to Nishizawa [9]. Let x ¼ e 2piz ; q j ¼ e 2pit j for zAC and t j AC À R ð0pjprÞ; and % q ¼ ðq 0 ; y y; q r Þ; % q À ð jÞ ¼ ðq 0 ; y;q j ; y; q r Þ;…”

Section: Introductionmentioning

confidence: 99%

“…The first formula follows from (9). Second formula of the theorem is similar to first one by letting o rþ1 ¼ þ1: &…”

mentioning

confidence: 96%

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The modular properties and the integral representations of the multiple elliptic gamma functions

Narukawa

2004

Advances in Mathematics

172

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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 modular property of the theta function is known as Jacobi's 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

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The modular properties and the integral representations of the multiple elliptic gamma functions

Narukawa

2004

Advances in Mathematics

172

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“…From Theorem 1, we remark that under the change of the variables, the q-multiple gamma function and the q-multiple sine function are equivalent to the q-shifted factorial and the multiple elliptic gamma function (see [17,18,23] for details), respectively.…”

Section: Theorem 1 We Havementioning

confidence: 99%

Multiple gamma functions, multiple sine functions, and Appell’s O-functions

Tanaka

2010

Ramanujan J

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Kurokawa introduced q-multiple gamma functions and q-multiple sine functions. We show that the Appell's O-function is expressed via the q-multiple gamma function. We also give some applications of this result. For example, we obtain a formula for the "Stirling modular form" and calculate special values of the q-multiple sine function. Moreover, we give some formulas of Eisenstein series and double cotangent functions and its generalization. Then the former gives an infinite product expression of the double sine function explicitly and a result of Kurokawa.

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“…are the first two of a sequence introduced by Nishizawa [22] of multiple elliptic gamma functions G n of n+2 variables obeying…”

Section: Multiple Gamma Functions and Bernoulli Polynomials The Funcmentioning

confidence: 99%

A gerbe for the elliptic gamma function

et al. 2008

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Abstract. The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three (it is a stack). Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve.

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An elliptic analogue of the multiple gamma function

Cited by 23 publications

References 14 publications

The modular properties and the integral representations of the multiple elliptic gamma functions

The modular properties and the integral representations of the multiple elliptic gamma functions

Multiple gamma functions, multiple sine functions, and Appell’s O-functions

A gerbe for the elliptic gamma function

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