On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

In this paper, we first give some results on monic generalized Hermite polynomials (GHP) {Hn(μ)(x)}n≥0, orthogonal with respect to the positive weight |x|2μe−x2,μ>−12,x∈R, which will lead to the formulation of the second-order spectralvectorial differential equation (SVDE) that the GHP satisfies....

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Bibliographic Details
Published in:Axioms 2022-07, Vol.11 (7), p.344
Main Authors: Atia, Mohamed Jalel, Benabdallah, Majed
Format: Article
Language:English
Subjects:
classical orthogonal polynomials
Differential equations
Eigenvalues
Hermite polynomials
Polynomials
Remakes & sequels
second order linear differential equation
semiclassical orthogonal polynomials
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Summary:In this paper, we first give some results on monic generalized Hermite polynomials (GHP) {Hn(μ)(x)}n≥0, orthogonal with respect to the positive weight |x|2μe−x2,μ>−12,x∈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 Jn(α,α+1)(x,μ), orthogonal with respect to the positive weight w(x,α;μ)=|x|−μ(1−x2)α(1−x), μ−1 on [−1,1].
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms11070344