On a novel class of polyanalytic Hermite polynomials

2019/02/26 by Benahmadi, Abdelhadi, Ghanmi, Allal
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.09945

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 L²--spectral theory of some special second order differential operators of Laplacian type acting on the L²--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 L²--gaussian Hilbert space on the strip.

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• Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials
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• The orthogonal complement of the Hilbert space associated to holomorphic Hermite polynomials
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