A basic class of symmetric orthogonal functions using the extended Sturm-Liouville theorem for symmetric functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:216

Article

Masjed-Jamei, Mohammad

关键词: extended Sturm-Liouville theorem for symmetric functions;    symmetric orthogonal functions;    Favard's theorem;    generalized ultraspherical polynomials;    generalized Hermite polynomials;    two sequences of finite classical symmetric orthogonal polynomials;   

DOI  :  10.1016/j.cam.2007.04.025

来源: Elsevier

【 摘 要 】

By using the extended Sturm-Liouville theorem for symmetric functions, we introduced a basic class of symmetric orthogonal polynomials (BCSOP) in a previous paper. The mentioned class satisfies a differential equation of the form x(2) (px(2) + q)Phi ''(n)(x) + x(rx(2) + s)Phi'(n) (x) - (n(r + (n-1)p)x(2) + (1-(-1)(n))s/2) Phi(n) (x) = 0 and contains four main sequences of symmetric orthogonal polynomials. in this paper, again by using the mentioned theorem, we introduce a basic class of symmetric orthogonal functions (BCSOF) as a generalization of BCSOP and obtain its standard properties. We show that the latter class satisfies the equation x(2) (px(2) + q) Phi(n)(x) + x(rx(2) + s)Phi'(n)(x) - (alpha(n)x(2) + (1-(-1)(n))beta/2)Phi(n) (x) = 0, in which beta is a free parameter and -alpha(n) denotes eigenvalues corresponding to BCSOF. We then consider four sub-classes of defined orthogonal functions class and study their properties in detail. Since BCSOF is a generalization of BCSOP for beta = s, the four mentioned sub-classes respectively generalize the generalized ultraspherical polynomials, generalized Hermite polynomials and two other finite sequences of symmetric polynomials, which were introduced in the previous work. (C) 2007 Elsevier B.V. All rights reserved.

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