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Analytic properties of complex Hermite polynomials


Author: Mourad E. H. Ismail
Journal: Trans. Amer. Math. Soc. 368 (2016), 1189-1210
MSC (2010): Primary 33C50, 33C70; Secondary 42C10, 05A40, 40B05
DOI: https://doi.org/10.1090/tran/6358
Published electronically: June 11, 2015
MathSciNet review: 3430361
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Abstract: We study the complex Hermite polynomials $ \{H_{m,n}(z, \bar z)\}$ 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 $ \{H_{m,n}(z,\bar z)\}$. Some inverse relations are also given. We give a two dimensional moment representation for $ H_{m,n}(z,\bar z)$ and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that $ \{H_{m,n}(z, \bar z)\}$ are the only bivariate orthogonal polynomials of Appell type.


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Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816 – and – Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
Email: mourad.eh.ismail@gmail.com

DOI: https://doi.org/10.1090/tran/6358
Keywords: 2$D$-Hermite polynomials, Poisson kernel, positivity of kernels, integral operators, multilinear generating functions, Kibble-Slepian formula, evaluation of integrals, zeros, Christoffel-Darboux identities, Appell polynomials
Received by editor(s): September 2, 2013
Received by editor(s) in revised form: December 22, 2013
Published electronically: June 11, 2015
Additional Notes: This research was supported by the NPST Program of King Saud University; project number 10-MAT1293-02 and the DSFP at King Saud University in Riyadh.
Dedicated: To my very good friend Paul Butzer on his $85$th birthday
Article copyright: © Copyright 2015 American Mathematical Society

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