$q$-Selberg Integrals and Koornwinder Polynomials

TL;DR

This paper generalizes the $q$-Selberg integral evaluation by incorporating Koornwinder polynomial norms, leading to new integral formulas and limits related to Mehta's integral, advancing the understanding of multivariate special functions.

Contribution

It introduces a generalized $q$-Selberg integral involving Koornwinder polynomial norms, providing new evaluation formulas and limit cases of classical integrals.

Findings

Proved a generalized $q$-Selberg integral formula.

Derived new limit cases related to Mehta's integral.

Connected Koornwinder polynomial norms with integral evaluations.

Abstract

We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.