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

We provide the explicit expression of first order $q$-difference system for the Jackson integral of symmetric Selberg type, which is generalized from the $q$-analog of contiguity 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 the explicit expression of the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called the interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials via Jackson integral representation of symmetric Selberg type, we compute the coefficient matrix.