The modular properties and the integral representations of the multiple elliptic gamma functions
Atsushi Narukawa
TL;DR
The paper generalizes Jacobi-type modular transformations to the hierarchy of multiple elliptic gamma functions $G_r(z|\underline{\tau})$ by exploiting dual infinite-product representations of associated multiple sine functions $S_r(z|\underline{\omega})$ and their integral forms. It develops integral representations for $S_r$ and defines a generalized $q$-polylogarithm $\mathrm{Li}_{r+2}(x;\underline{q})$, establishing the key connection $(x;\underline{q})^{(r)}_\infty = \exp(-\mathrm{Li}_{r+2}(x;\underline{q}))$, which underpins product representations and modular identities. A central result is the modular identity $\prod_{k=1}^r G_{r-2}(\frac{z}{\omega_k}|(\frac{\omega_1}{\omega_k},\dots)) = \exp(-\frac{2\pi i}{r!} B_{rr}(z|\underline{\omega}))$, generalizing Felder–Varchenko-type transformations. The paper also shows that $G_r$ can be represented either by a contour integral or as an infinite product of $S_{r+1}$ functions, clarifying the structural link between $G_r$ and the family of $S_r$ and providing tools for analytic and spectral applications.
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 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.