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 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.

Hypergeometric solutions for variants of the $q$-hypergeometric equation

“…A q-difference system for the Jackson integral of Selberg type, which contains the system for the Jordan-Pochhammer type, was obtained in [19]. In [11], a q-difference system for the Jackson integral of Selbarg type was also discussed, and above results were summarized. Hence, for more details of the Jackson integral of Jordan-Pochhammer type, see [11,17,18,19].…”

“…In [11], a q-difference system for the Jackson integral of Selbarg type was also discussed, and above results were summarized. Hence, for more details of the Jackson integral of Jordan-Pochhammer type, see [11,17,18,19]. In this paper, we derive a q-difference equation for the Jackson integral of Jordan-Pochhammer type by integrating the equation that the integrand satisfies.…”

Connection Problem for an Extension of q-Hypergeometric Systems

“…A q-difference system for the Jackson integral of Selberg type, which contains the system for the Jordan-Pochhammer type, was obtained by Mimachi [26]. In [14], a q-difference system for the Jackson integral of Selbarg type was also discussed, and the above results were summarized. Hence, for more details on the Jackson integral of Jordan-Pochhammer type, see [14,21,25,26].…”

“…In [14], a q-difference system for the Jackson integral of Selbarg type was also discussed, and the above results were summarized. Hence, for more details on the Jackson integral of Jordan-Pochhammer type, see [14,21,25,26].…”

“…is an active area of research for special functions, combinatorics, mathematical physics and orthogonal polynomials. For details and histories, see [14,15]. To consider a "Riemann-Papperitz type extension of the q-Selberg integral" is interesting and important.…”