# On a novel class of polyanalytic Hermite polynomials

## TL;DR

This paper introduces a new class of orthogonal polyanalytic Hermite polynomials, exploring their properties, connections to differential operators, and applications in spectral theory and automorphic functions.

## Contribution

It develops algebraic and analytic properties of these polynomials and links them to spectral theory and automorphic functions, providing new tools and insights.

## Key findings

- Derived operational formulas and recurrence relations.
- Established orthogonality identities and integral representations.
- Connected polyanalytic polynomials to spectral theory of differential operators.

## Abstract

We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. We establish their connection and rule in describing the $L^2$--spectral theory of some special second order differential operators of Laplacian type acting on the $L^2$--gaussian Hilbert space on the whole complex plane. We will also show their importance in the theory of the so-called rank--one automorphic functions on the complex plane. In fact, a variant subclass leads to an orthogonal basis of the corresponding $L^2$--gaussian Hilbert space on the strip.