  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.
