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

Abstract

We provide an explicit expression for the first-order 𝑞-difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of the 𝑞-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system, we use a set of symmetric polynomials introduced by Matsuo in his study of the 𝑞-KZ equation. Our main result is an explicit expression for the coefficient matrix of the 𝑞-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials, we compute the coefficient matrix.