Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

TL;DR

This paper derives generating functions for products of Laguerre 2D and Hermite 2D polynomials, extending classical formulas and providing tools for Gaussian convolutions and polynomial calculations in mathematical physics.

Contribution

It introduces new generating functions for products of Laguerre 2D and Hermite 2D polynomials, including mixed products, and develops operator identities for Gaussian polynomial convolutions.

Findings

Derived bilinear generating functions for Laguerre 2D polynomials.

Extended Mehler's formula to products of Hermite and Laguerre 2D polynomials.

Provided operator identities for Gaussian-polynomial convolutions.

Abstract

The bilinear generating function for products of two Laguerre 2D polynomials Lm;n(z; z0) with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of SU(1; 1) operator disentanglement some operator identities are derived in an appendix. They allow to calculate convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions. Keywords: Laguerre and Hermite polynomials, Laguerre 2D polynomials, Jacobi polynomials, Mehler formula, SU(1; 1) operator disentanglement, Gaussian convolutions.