Multivariate holomorphic Hermite polynomials
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Multivariate holomorphic Hermite polynomials Mourad E. H. Ismail1 · Plamen Simeonov2 Received: 13 September 2018 / Accepted: 11 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Multivariate holomorphic Hermite polynomials · Generating function · Orthogonality · Hilbert space · Multivariate Itô polynomials · Kibble–Slepian formula Mathematics Subject Classification 33C45 · 42C05 · 33C50
1 Introduction There has been a number of recent studies devoted to bivariate and multivariate orthogonal polynomials related to and generalizing the Hermite polynomials. Such polynomials are for example Itô’s polynomials in one complex variable [22] (the 1-D Itô polynomials), whose combinatorial and analytic properties were investigated in
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Plamen Simeonov [email protected] Mourad E. H. Ismail [email protected]
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Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
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Department of Mathematics and Statistics, University of Houston-Downtown, Houston, TX 77002, USA
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M. E. H. Ismail, P. Simeonov
[16,17,19–21], and the real Hermite polynomials in n real variables first introduced in the 19th century by Hermite [3], [7, Chapter 12], [14], and recently studied in [18]. Combinatorial interpretations of integrals of products of Itô and multivariate real Hermite polynomials using generating functions for these integrals were derived in [17,18], extending previous results by Foata [8,9], and Foata and Garsia [10]. A Kibble–Slepian formula for the univariate Itô polynomials was given in [16]. A Kibble– Slepian formula for multivariate real Hermite polynomials that generalizes the classical Kibble–Slepian formula [15,23,25,27] was established in [18]. In this paper we will introduce holomorphic Hermite polynomials in n complex variables (n-D holomorphic Hermite polynomials), and Itô-type polynomials in n complex variables which we will also call n-D Itô polynomials. We will then derive generating functions, describe cases when these polynomials are orthogonal, and establish a number of other properties and identities for these two types of polynomials, including a Kibble–Slepian and a Mehler-type formula for the n-D Itô polynomials. These results extend
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