Differential Algebraicity of the Multiple Elliptic Gamma Function for a Rational Period

Abstract

The multiple elliptic gamma function includes q-analogues of the multiple sine function, introduced by Kurokawa. Thus it is natural to consider whether properties of the multiple sine function can be extended to that of the multiple elliptic gamma function. Kurokawa and Wakayama showed that, when the period is "rational", the multiple sine function satisfies an algebraic differential equation. In this paper, we investigate the differential algebraicity of the multiple elliptic gamma function. We show that the multiple elliptic gamma function G_r(z, τ) satisfies an algebraic differential equation when the period τ is "rational".