The $q$-Dixon--Anderson integral and multi-dimensional $_1\psi_1$ summations

TL;DR

This paper explores the deep connections between the $q$-Dixon--Anderson integral and multi-dimensional $_1 ext{ψ}_1$ summations, revealing their common origin in $q$-difference equations and providing new insights into their structure.

Contribution

It demonstrates that both the $q$-Dixon--Anderson integral and multi-dimensional $_1 ext{ψ}_1$ summations originate from rank-one $q$-difference equations, unifying their understanding.

Findings

Both integrals and summations are governed by a common $q$-difference equation.

The analysis employs concepts of truncation, regularization, and connection formulae.

The results clarify the structural relationship between these $q$-series and integrals.

Abstract

The Dixon--Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The $_1\psi_1$ summation is a bilateral generalization of the $q$-binomial theorem. It is shown that a $q$-generalization of the Dixon--Anderson integral, due to Evans, and multi-dimensional generalizations of the $_1\psi_1$ summation, due to Milne and Gustafson, can be viewed as having a common origin in the theory of $q$-difference equations as expounded by Aomoto. Each is shown to be determined by a $q$-difference equation of rank one, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.