On a Family of Integrals that extend the Askey-Wilson Integral

Abstract

We study a family of integrals parameterised by $N=2,3,\dots$ generalising the Askey-Wilson integral $N=2$ which has arisen in the theory of $q$-analogs of monodromy preserving deformations of linear differential systems and in theory of the Baxter $Q$ operator for the $XXZ$ open quantum spin chain. These integrals are particular examples of moments defined by weights generalising the Askey-Wilson weight and we show the integrals are characterised by various $(N-1)$-th order linear $q$-difference equations which we construct. In addition we demonstrate that these integrals can be evaluated as a finite sum of $(N-1)$ $B{C}_1$-type Jackson integrals or ${}_{2N+2}{\phi }_{2N+1}$ basic hypergeometric functions.

arXiv comment26 pages

References in corpus (4)

• Properties of generalized univariate hypergeometric functions
• Bethe Ansatz Equations of XXZ Model and q-Sturm-Liouville Problems
• Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
• Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy