Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

Abstract

We consider the continuous $q$-Jacobi polynomials $\{P_n^{(\alpha,\beta)}(\cdot|q)\}_{n=0}^{\infty}$, extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the $q$-Sobolev character of the non-standard orthogonality for the corresponding family.