Kibble-Slepian Formula and Generating Functions for 2D Polynomials

We prove a generalization of the Kibble--Slepian formula (for Hermite polynomials) and its unitary analogue involving the $2$D Hermite polynomials recently proved in \cite{Ism4}. We derive integral representations for the $2$D Hermite polynomials which are of independent interest. Several new generating functions for $2$D $q$-Hermite polynomials will also be given.