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

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 $(N-1)$ $BC_{1}$-type Jackson integrals or ${}_{2N+2}\varphi_{2N+1}$ basic hypergeometric functions.