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Article

Monomiality and a New Family of Hermite Polynomials

by
Giuseppe Dattoli
1,† and
Silvia Licciardi
2,*,†
1
ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy
2
Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2023, 15(6), 1254; https://doi.org/10.3390/sym15061254
Submission received: 21 April 2023 / Revised: 29 May 2023 / Accepted: 30 May 2023 / Published: 13 June 2023
(This article belongs to the Special Issue Theory and Applications of Special Functions II)

Abstract

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.
Keywords: special functions 33C52, 33C65, 33C99, 33B10, 33B15; Hermite polynomials 33C45; operators theory 44A99, 47B99, 47A62 special functions 33C52, 33C65, 33C99, 33B10, 33B15; Hermite polynomials 33C45; operators theory 44A99, 47B99, 47A62

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MDPI and ACS Style

Dattoli, G.; Licciardi, S. Monomiality and a New Family of Hermite Polynomials. Symmetry 2023, 15, 1254. https://doi.org/10.3390/sym15061254

AMA Style

Dattoli G, Licciardi S. Monomiality and a New Family of Hermite Polynomials. Symmetry. 2023; 15(6):1254. https://doi.org/10.3390/sym15061254

Chicago/Turabian Style

Dattoli, Giuseppe, and Silvia Licciardi. 2023. "Monomiality and a New Family of Hermite Polynomials" Symmetry 15, no. 6: 1254. https://doi.org/10.3390/sym15061254

APA Style

Dattoli, G., & Licciardi, S. (2023). Monomiality and a New Family of Hermite Polynomials. Symmetry, 15(6), 1254. https://doi.org/10.3390/sym15061254

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