We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an $L^2$ 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 $e^{ ixy}$ in Jacobi polynomials of argument $x$. We also outline a general procedure of finding integral representations for inverses of linear operators.
Categories: math.CA