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

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.

Connection problem in holonomic q-difference system associated with a Jackson integral of Jordan-Pochhammer type

Fix a complex number q with |q| < 1. Let T1…, Tn be n-commuting q-difference operators defined byfor a function f(x), x = (x1,…,xn) ε (C*)n. Consider a system of linear q-difference equations in several variables for a matrix valued function on (C*)n as follows: