On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

Abstract

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, α>−1 on [−1,1].