A bilateral extension of the q-Selberg integral

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.