We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an L2 space weighted by the weight function of the continuous q-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a q-analog of the expansion of eixy in Jacobi polynomials of argument x. We also outline a general procedure of finding integral repre...

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bDq). Using this operator, we obtain extensi...