Multivariate holomorphic Hermite polynomials
Hermite polynomials was introduced by Hermite in the late 19th century. Recently there have been a number of studies devoted to bivariate and multivariate orthogonal polynomials related to Hermite polynomials and generalizing them. Such polynomials are, for instance, the real Hermite polynomials in \(n\) real variables and the Itô's polynomials in one complex variable. The last ones were introduced by Itô to study complex Wiener integrals and later used to analyze the so-called coherent states.The Itô polynomials are known in physics as the Feynman-Schwinger basis. In this paper the authors introduce holomorphic Hermite polynomials in \(n\) complex variables (\(n\)-D holomorphic Hermite polynomials), and Itô type polynomials in \(n\) complex variables, (\(n\)-D Itô polynomials). These two families of polynomials generalize the Hermite polynomials in \(n\) real variables. The authors 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 holomorphic Hermite polynomials. They also construct a reproducing kernel Hilbert space associated with a class of orthogonal Hermite polynomials and provide the orthonormal basis and the reproducing kernel for this Hilbert space. Along of the paper the authors give a general approach to study orthogonal \(n\)-D holomorphic Hermite polynomials. They obtain a variety of analytic properties for the \(n\)-D holomorphic Hermite polynomials and establish other properties related to the Itô polynomials analogous to those of Hermite polynomials.
multivariate holomorphic Hermite polynomials
scientific article; zbMATH DE number 7345240
scientific article