Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 113, 31 pages      arXiv:1910.08393      https://doi.org/10.3842/SIGMA.2020.113
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

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

Masahiko Ito
Department of Mathematical Sciences, University of the Ryukyus, Okinawa 903-0213, Japan

Received April 29, 2020, in final form October 29, 2020; Published online November 08, 2020

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.

Key words: $q$-difference equations; Selberg type integral; contiguous relations; Gauss decomposition.

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