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

Alfred W&uuml; nsche

doi:10.4236/am.2015.612188

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Applied Mathematics > Vol.6 No.12, November 2015

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

Alfred Wünsche

Institut für Physik, Nichtklassische Strahlung, Humboldt-Universit?t, Berlin, Germany.

DOI: 10.4236/am.2015.612188 PDF HTML XML 3,040 Downloads 3,973 Views Citations

Abstract

The bilinear generating function for products of two Laguerre 2D polynomials

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

operator disentanglement some operator identities are derived in an appendix. They allow calculating 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

Share and Cite:

Wünsche, A. (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. Applied Mathematics, 6, 2142-2168. doi: 10.4236/am.2015.612188.

Conflicts of Interest

The authors declare no conflicts of interest.

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