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

We study a family of integrals parameterised by N = 2, 3, . . . 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) BC1-type Jackson integrals or 2N+2φ2N+1 basic hypergeometric functions.