A Combinatorial Approach To The 2D-Hermite And 2D-Laguerre Polynomials

Keywords

2D-Hermite polynomials; 2D-Laguerre polynomials; Elementary symmetric functions; Inequalities; Kibble-Slepian formula; Linearization coefficients; Positivity; Shifted Laguerre polynomials

Abstract

The first author has recently proved a Kibble-Slepian type formula for the 2D-Hermite polynomials {Hm,n(z, z¯)} which extends the Poisson kernel for these polynomials. We provide a combinatorial proof of a closely related formula. The combinatorial structures involved are the so-called m-involutionary ℓ-graphs. They are enumerated in two different manners: first globally, then as the exponential of their connected components. We also give a combinatorial model for the 2D-Laguerre polynomials and study their linearization coefficients.

Publication Date

1-1-2015

Publication Title

Advances in Applied Mathematics

Volume

64

Issue

1

Number of Pages

70-88

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.aam.2014.12.002

Socpus ID

84922751846 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84922751846

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