A bilateral extension of the $q$-Selberg integral

TL;DR

This paper extends the $q$-Selberg integral to a multi-dimensional bilateral $q$-series using advanced techniques like truncation, regularization, and $q$-difference equations, providing new evaluations and generalizations.

Contribution

It introduces a bilateral extension of the $q$-Selberg integral and derives a new $q$-difference equation framework using shifted symmetric polynomials.

Findings

Derived an infinite product evaluation of the bilateral $q$-series.

Reclaimed and generalized known $q$-Selberg integral evaluations.

Provided a new perspective on the $q$-Selberg integral through Aomoto's method.

Abstract

A multi-dimensional bilateral $q$-series extending the $q$-Selberg integral is studied using concepts of truncation, regularization and connection formulae. Following Aomoto's method, which involves regarding the $q$-series as a solution of a $q$-difference equation fixed by its asymptotic behavior, an infinite product evaluation is obtained. The $q$-difference equation is derived applying the shifted symmetric polynomials introduced by Knop and Sahi. As a special case of the infinite product formula, Askey--Evans's $q$-Selberg integral evaluation and its generalization by Tarasov--Varchenko and Stokman is reclaimed, and an explanation in the context of Aomoto's setting is thus provided.