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Multivariate holomorphic Hermite polynomials

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Abstract

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Itô polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Itô polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Itô polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Itô polynomials are established.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Mourad E. H. Ismail

  2. Department of Mathematics and Statistics, University of Houston-Downtown, Houston, TX, 77002, USA

    Plamen Simeonov

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  1. Mourad E. H. Ismail
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  2. Plamen Simeonov
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Correspondence to Plamen Simeonov.

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Ismail, M.E.H., Simeonov, P. Multivariate holomorphic Hermite polynomials. Ramanujan J 53, 357–387 (2020). https://doi.org/10.1007/s11139-020-00312-8

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  • DOI: https://doi.org/10.1007/s11139-020-00312-8

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