On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

Atia, Mohamed Jalel; Benabdallah, Majed

doi:10.3390/axioms11070344

Open AccessArticle

On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

by

Mohamed Jalel Atia

^1,*

and

Majed Benabdallah

²

¹

Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia

²

Laboratory of Mathematics and Applications (LR17ES11), Faculty of Sciences of Gabes, University of Gabès, Gabès 6072, Tunisia

^*

Author to whom correspondence should be addressed.

Axioms 2022, 11(7), 344; https://doi.org/10.3390/axioms11070344

Submission received: 24 May 2022 / Revised: 3 July 2022 / Accepted: 11 July 2022 / Published: 19 July 2022

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)

Download Review Reports Versions Notes

Abstract

In this paper, we first give some results on monic generalized Hermite polynomials (GHP)

{H_{n}^{(μ)} (x)}_{n \geq 0}

, orthogonal with respect to the positive weight

{| x |}^{2 μ} e^{- x^{2}}, μ > - \frac{1}{2}, x \in R

, which will lead to the formulation of the second-order spectralvectorial differential equation (SVDE) that the GHP satisfies. This SVDE differs from the one given in G. Szego (problem 25. p. 380), which is a pseudo-spectral equation. Second, we give the SVDE, as conjecture, satisfied by the generalized Jacobi polynomials

J_{n}^{(α, α + 1)} (x, μ)

, orthogonal with respect to the positive weight

w (x, α; μ) = {| x |}^{- μ} {(1 - x^{2})}^{α} (1 - x),

μ < 1,

α > - 1

on

[- 1, 1]

.

Keywords: classical orthogonal polynomials; semiclassical orthogonal polynomials; second order linear differential equation

Share and Cite

MDPI and ACS Style

Atia, M.J.; Benabdallah, M. On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials. Axioms 2022, 11, 344. https://doi.org/10.3390/axioms11070344

AMA Style

Atia MJ, Benabdallah M. On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials. Axioms. 2022; 11(7):344. https://doi.org/10.3390/axioms11070344

Chicago/Turabian Style

Atia, Mohamed Jalel, and Majed Benabdallah. 2022. "On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials" Axioms 11, no. 7: 344. https://doi.org/10.3390/axioms11070344

APA Style

Atia, M. J., & Benabdallah, M. (2022). On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials. Axioms, 11(7), 344. https://doi.org/10.3390/axioms11070344

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI