On a Novel Class of Polyanalytic Hermite Polynomials

Abstract

We discuss some algebraic and analytic properties of a general 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 $${\mathbb {C}}/{\mathbb {Z}}$$.