Holomorphic Hermite polynomials in two variables

TL;DR

This paper introduces and explores holomorphic Hermite polynomials in two variables, analyzing their algebraic and analytic properties, and connecting them to classical spaces like the Bargmann space through limit behaviors.

Contribution

The paper is the first to investigate holomorphic Hermite polynomials in two variables, extending their algebraic and analytic understanding and relating them to existing polynomial frameworks.

Findings

Developed algebraic properties of the new polynomials.

Analyzed their analytic features based on non-rotational orthogonality.

Established connections to Bargmann space and classical Hermite polynomials through limits.

Abstract

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$.