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

TL;DR

This paper derives an explicit first-order $q$-difference system for the Jackson integral of symmetric Selberg type, generalizing $q$-analog of hypergeometric relations using symmetric polynomials.

Contribution

It provides an explicit expression for the coefficient matrix of the $q$-difference system using Gauss matrix decomposition and introduces interpolation polynomials for computation.

Findings

Explicit $q$-difference system for Jackson integral of symmetric Selberg type.

Coefficient matrix expressed via Gauss matrix decomposition.

Introduction of interpolation polynomials for system computation.

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 $q$-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the 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.