Kernel identities for van Diejen's $q$-difference operators and transformation formulas for multiple basic hypergeometric series

TL;DR

This paper establishes kernel identities linking van Diejen's $q$-difference operators with hypergeometric series transformations, and introduces new commuting operators with Koornwinder polynomials as eigenfunctions.

Contribution

It provides new kernel identities for type $BC$ $q$-difference operators and constructs a novel family of commuting operators with Koornwinder polynomials.

Findings

Kernel function of Cauchy type intertwines van Diejen's operators

Transformation formulas for multiple basic hypergeometric series of type $BC$

New family of commuting $q$-difference operators with Koornwinder polynomials as eigenfunctions

Abstract

In this paper, we show that the kernel function of Cauchy type for type $BC$ intertwines the commuting family of van Diejen's $q$-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type $BC$. We also construct a new infinite family of commuting $q$-difference operators for which the Koornwinder polynomials are joint eigenfunctions.