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Complex Hermite polynomials: Their combinatorics and integral operators

Authors: Mourad E. H. Ismail and Plamen Simeonov
Journal: Proc. Amer. Math. Soc. 143 (2015), 1397-1410
MSC (2010): Primary 05A15, 05A18, 33C45, 45P05; Secondary 42A65
Published electronically: December 9, 2014
MathSciNet review: 3314055
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Abstract: We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponential weights in the complex plane. We also construct four integral operators of which the first type of complex Hermite polynomials are eigenfunctions and we identify the corresponding eigenvalues. We prove that the products of these complex Hermite polynomials are complete in certain $L_2$-spaces.

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Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816 – and – King Saud University, Riyadh, Saudi Arabia
MR Author ID: 91855

Plamen Simeonov
Affiliation: Department of Mathematics and Statistics, University of Houston-Downtown, Houston, Texas 77002

Keywords: Complex Hermite polynomials, matchings of multisets, orthogonality, combinatorics of linearization of products, eigenvalues, eigenfunctions, integral operators, completeness
Received by editor(s): January 25, 2013
Received by editor(s) in revised form: May 31, 2013, and July 18, 2013
Published electronically: December 9, 2014
Additional Notes: The first author’s research was supported by the NPST Program of King Saud University; project number 10-MAT1293-02 and King Saud University in Riyadh and by the Research Grants Council of Hong Kong grant # CityU 1014111.
Communicated by: Jim Haglund
Article copyright: © Copyright 2014 American Mathematical Society