Kibble-Slepian Formula and Generating Functions for 2$D$ Polynomials

TL;DR

This paper generalizes the Kibble-Slepian formula for 2D Hermite polynomials, introduces new generating functions, and provides integral representations, advancing theoretical understanding of multivariate polynomial identities.

Contribution

It extends the Kibble-Slepian formula to 2D Hermite polynomials and derives new generating functions and integral representations.

Findings

Generalized Kibble-Slepian formula for 2D Hermite polynomials

Derived integral representations for 2D Hermite polynomials

Introduced new generating functions for 2D q-Hermite polynomials

Abstract

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.