More basic hypergeometric limits of the elliptic hypergeometric beta integral

TL;DR

This paper explores new limits of the elliptic hypergeometric beta integral as p approaches zero along specific sequences, revealing connections to bilateral basic hypergeometric series like the {}_6 ext{ extpsi}_6 evaluation.

Contribution

It extends previous work by allowing p to tend to zero along geometric sequences with specific parameter restrictions, enabling the derivation of new hypergeometric limits.

Findings

Derived new limits involving bilateral basic hypergeometric series.

Obtained evaluation formulas for {}_6 ext{ extpsi}_6 series.

Expanded the class of hypergeometric limits accessible from elliptic integrals.

Abstract

In this article we continue the work from arXiv:0902.0621. In that article Eric Rains and the present author considered the limits of the elliptic beta integral as p->0 while the parameters t_r have a p-dependence of the form t_r=u_rp^{\alpha_r} (for fixed u_r and certain real numbers \alpha_r). In this article we again consider such limits, but now we let p->0 along a geometric sequence p=xq^{sk} (for some integer s, while k -> \infty), and only allow \alpha_r\in 2/s Z. These choices allow us to take many more limits. In particular we now also obtain bilateral basic hypergeometric series as possible limits, such as the evaluation formula for a very well poised {}_6\psi_6.