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.

Connection Formula for the Jackson Integral of Type A_n and Elliptic Lagrange Interpolation

“…The formula (1.9) was discovered by Askey [5] and proved by Habsieger [11], Kadell [16], Evans [6] and Kaneko [19] in the case where τ is a positive integer, while (1.9) for general complex τ was given by Aomoto [1]. See [13] for further details. Other closely related and relevant works are [17,18,33].…”

q-Difference Systems for the Jackson Integral of Symmetric Selberg Type

“…After defining some basic terminology in Section 2, we characterize in Section 3 Matsuo's polynomials by their vanishing property (Proposition 3.1), and define a family of symmetric polynomials of higher degree, which includes Matsuo's polynomials. We call such polynomials the interpolation polynomials, which are inspired from Aomoto's method [2, Section 8], [3], which is a technique to obtain difference equations for the Selberg integrals (see also [18] for a q-analogue of Aomoto's method). We state several vanishing properties for the interpolation polynomials, which are used in subsequent sections.…”

“…When m = 1 the Jackson integral of symmetric Selberg type is equivalent to the q-Selberg integral defined by Askey [11] and proved by others, see [6,13,16,20] for instance. See also recent references [15,Section 2.3] and [18]. q-Selberg integral is a very active area of research with important connections to special functions, combinatorics, mathematical physics and orthogonal polynomials (see [1,23,24,25,32,35] and [17,Section 5]).…”

q-Selberg Integrals and Koornwinder Polynomials

“…Aomoto [2] proposed and evaluated q-Selberg type integrals attached to any reduced root systems. Let us mention some recent relevant papers: other proofs of the Habsieger-Kadell's formula [11,20,36] and variations and extensions [16,34,35]. To state the formula we require some definitions.…”