A right inverse of the Askey-Wilson operator

We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on $L^2$ with weight $(1-x^2)^{-1/2}$. The kernel of this integral operator is $\vartheta'_4/\vartheta_4$ and is the Riemann mapping function that maps the open unit disc conformally onto the interior of an ellipse.