On a family of integrals that extend the Askey–Wilson integral

We study a family of integrals parameterised by N=2,3,…N=2,3,… generalising the Askey–Wilson integral N=2N=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)(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)(N−1)BC1BC1-type Jackson integrals or φ2N+12N+2 basic hypergeometric functions.