Infinite elliptic hypergeometric series: convergence and difference equations

TL;DR

This paper studies the convergence properties of infinite elliptic hypergeometric series, deriving difference equations and establishing conditions under which these series converge, extending classical criteria to the elliptic case.

Contribution

It introduces finite difference equations of infinite order for theta hypergeometric series and extends convergence criteria from $q$-hypergeometric to elliptic hypergeometric series.

Findings

Derived finite difference equations for theta hypergeometric series.

Identified parameter constraints for convergence of elliptic hypergeometric series.

Proved convergence of ${}_{r+1}V_r$ elliptic hypergeometric series under specific conditions.

Abstract

We derive finite difference equations of infinite order for theta hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion on the convergence of $q$-hypergeometric series for $|q|=1, \, q^n\neq 1$, to the elliptic level and prove convergence of the infinite ${}_{r+1}V_r$ very-well poised elliptic hypergeometric series for restricted values of $q$.