Ramanujan's $_1\psi_1$ summation theorem --- perspective, announcement of bilateral $q$-Dixon--Anderson and $q$-Selberg integral extensions, and context

TL;DR

This paper explores Ramanujan's $_1 heta_1$ summation theorem using $q$-integral and $q$-difference methods, introduces new multidimensional bilateral Jackson integral extensions of classical integrals, and contextualizes these within existing literature.

Contribution

It presents novel multidimensional bilateral Jackson integral extensions of the Dixon--Anderson and Selberg integrals, connecting them to the $_1 heta_1$ summation theorem and previous product formulas.

Findings

Identification of bilateral Jackson integral generalizations for multidimensional cases.

New results on $q$-extensions of Dixon--Anderson and Selberg integrals.

Contextualization of these results within existing $q$-integral literature.

Abstract

The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised $_6\psi_6$ summation. Bilateral Jackson integral generalizations of the Dixon--Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_1\psi_1$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulas for Jackson integrals relating to other roots systems obtained from the same perspective.