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On a Novel Class of Polyanalytic Hermite Polynomials

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

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  1. Analysis, P.D.E. and Spectral Geometry, Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. Box 1014, Rabat, Morocco

    Abdelhadi Benahmadi & Allal Ghanmi

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  1. Abdelhadi Benahmadi
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  2. Allal Ghanmi
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Correspondence to Allal Ghanmi.

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Benahmadi, A., Ghanmi, A. On a Novel Class of Polyanalytic Hermite Polynomials. Results Math 74, 186 (2019). https://doi.org/10.1007/s00025-019-1110-z

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