Bivariate poly-analytic Hermite polynomials

TL;DR

This paper introduces a new class of bivariate poly-analytic Hermite polynomials, exploring their properties, generating functions, and applications in integral transforms and spectral theory of magnetic Laplacians.

Contribution

It presents the first detailed study of bivariate poly-analytic Hermite polynomials, including their orthogonality, operational forms, and connections to Fourier-Wigner transforms.

Findings

They form an orthogonal basis in the Hilbert space on two-complex space.

Derived recurrence relations, generating functions, and differential equations.

Applications to integral transforms and spectral analysis of magnetic Laplacians.

Abstract

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner's property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed.