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Results in Mathematics

, 74:186 | Cite as

On a Novel Class of Polyanalytic Hermite Polynomials

  • Abdelhadi Benahmadi
  • Allal GhanmiEmail author
Article
  • 69 Downloads

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 \({\mathbb {C}}/{\mathbb {Z}}\).

Keywords

Holomorphic Hermite polynomial polyanalytic complex Hermite polynomial generating function orthogonality relation integral representation polyanalytic functions rank-one autmorphic theta functions 

Mathematics Subject Classification

Primary 33E05 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Analysis, P.D.E. and Spectral Geometry, Department of Mathematics, Faculty of SciencesMohammed V UniversityRabatMorocco

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