Elliptic hypergeometric functions: integrals versus series

TL;DR

This paper explores the relationship between elliptic hypergeometric integrals and series, analyzing their representations, convergence properties, and asymptotic behavior of related sums.

Contribution

It provides a novel representation of the elliptic beta integral as a combination of elliptic hypergeometric series and discusses convergence and asymptotics.

Findings

Representation of the elliptic beta integral as a series combination

Conditions for convergence of the series combination

Asymptotic analysis of the Frenkel--Turaev sum as n approaches infinity

Abstract

The univariate elliptic beta integral is represented as a bilinear combination of infinite $_{10}V_9$ very-well-poised elliptic hypergeometric series representing the sum of residues of the integrand poles. Convergence of this combination of series for some particular choice of parameters is discussed. Additionally, the asymptotics of the Frenkel--Turaev sum for a terminating $_{10}V_9$ series is considered when the termination parameter $n$ goes to infinity.