On a Novel Class of Polyanalytic Hermite Polynomials

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.

The orthogonal complement of the Hilbert space associated to holomorphic Hermite polynomials

“…Remark 3.3. The new class of functions in (3.1) generalizes the one studied in [2] and the previous theorem provide an integral representation of the special functions ψ s m,n (z, z). Moreover, it is closely connected to the polynomials…”

“…For fixed a > 0, b ∈ R and c ∈ C, we define I a,b n (z, z|c) to be the class of polyanalytic polynomials in [2], I a,b n (z, z|c) := (−1) n e a|z| 2 −bz 2 −cz ∂ n z e −a|z| 2 +bz 2 +cz . Theorem 3.7.…”