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

Details 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

2003-06-10

arxiv.org/abs/math/0306164v3

Adv. in Math. 189 (2) (2004) 247-267

Mathematics

Quantum Algebra

20 pages, LaTeX, To appear in "Adv. in Math."

Scientific paper

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 product. 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.

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