On generalized Hermite polynomials

Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori; Waleed Mohamed Abd-Elhameed; Omar Mazen Alqubori

doi:10.3934/math.20241556

AIMS Mathematics

2024, Volume 9, Issue 11: 32463-32490. doi: 10.3934/math.20241556

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On generalized Hermite polynomials

Waleed Mohamed Abd-Elhameed ^{1,2
,
,},
Omar Mazen Alqubori ^{1
,
,}

1.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received: 06 September 2024 Revised: 19 October 2024 Accepted: 04 November 2024 Published: 15 November 2024
MSC : 33C45, 42C05, 33C05

This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs.
- generalized Hermite polynomials,
- hypergeometric functions,
- connection,
- linearization formulas,
- high-order derivatives
Citation: Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori. On generalized Hermite polynomials[J]. AIMS Mathematics, 2024, 9(11): 32463-32490. doi: 10.3934/math.20241556

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Abstract

This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs.

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