Analytic Properties Of Complex Hermite Polynomials

Keywords

2D-Hermite polynomials; Appell polynomials; Christoffel-Darboux identities; Evaluation of integrals; Integral operators; Kibble-Slepian formula; Multilinear generating functions; Poisson kernel; Positivity of kernels; Zeros

Abstract

We study the complex Hermite polynomials {(formula presented)} in some detail, establish operational formulas for them and prove a Kibble-Slepian type formula, which extends the Poisson kernel for these polynomials. Positivity of the associated kernels is discussed. We also give an infinite family of integral operators whose eigenfunctions are {(formula presented)}. Some inverse relations are also given. We give a two dimensional moment representation for (formula presented) and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that {(formula presented)} are the only bivariate orthogonal polynomials of Appell type.

Publication Date

2-1-2016

Publication Title

Transactions of the American Mathematical Society

Volume

368

Issue

2

Number of Pages

1189-1210

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/tran/6358

Socpus ID

84952065269 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84952065269

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