Multivariate holomorphic Hermite polynomials

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 Ito polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Ito polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Ito polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Ito polynomials are established.