A study of elliptic gamma function and allies

TL;DR

This paper explores the properties of the elliptic gamma function, its connections to elliptic dilogarithms and Eisenstein series, and extends these results to elliptic polylogarithms, enriching the understanding of special functions in complex analysis.

Contribution

It introduces new links between the elliptic gamma function, elliptic dilogarithms, Eisenstein series, and elliptic polylogarithms, expanding the theoretical framework of special functions.

Findings

Established connections between elliptic gamma function and elliptic dilogarithm.

Linked elliptic gamma function to non-holomorphic Eisenstein series.

Extended results to elliptic polylogarithms using Zagier's formulas.

Abstract

We study analytic and arithmetic properties of the elliptic gamma function $$ \prod_{m,n=0}^\infty\frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \qquad |q|,|p|<1, $$ in the regime $p=q$; in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eistenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.