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

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. *

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

“…A similar identity was used in [Rui09b] and [KNS09] to prove the kernel function identity for the the van Diejen operator (1). It is also possible that (24) can be obtained from the determinant identities of type BC [Mas13,KMN16]. As this is a key Lemma for our main results, we proceed to give a short proof using Liouville's Theorem.…”

An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A

“…Also, an earlier version of the present paper, written around 2012, has been circulated among some researchers. For these reasons, some of the results in this paper are already cited in several studies [2,3,8,9] with reference to a private communication or to a paper in preparation.…”

Duality Transformation Formulas for Multiple Elliptic Hypergeometric Series of Type BC

“…It would be an important problem to clarify the relation between our results for multiple series and summation and transformation formulas for elliptic hypergeometric integrals of type BC n as discussed by van Diejen -Spiridonov [2] and Rains [11]. In a previous work, the second author [10] derived a duality transformation formula for basic hypergeometric series of type BC n from a kernel identity for the commuting family of van Diejen q-difference operators. We expect that our transformation formulas can also be understood in a context similar to that of [10].…”

“…In a previous work, the second author [10] derived a duality transformation formula for basic hypergeometric series of type BC n from a kernel identity for the commuting family of van Diejen q-difference operators. We expect that our transformation formulas can also be understood in a context similar to that of [10].…”