A basic class of symmetric orthogonal functions with six free parameters
type: Item
instance of: scholarly article
title: A basic class of symmetric orthogonal functions with six free parameters
zbMATH Open document ID: 1195.33068
author: Mohammad Masjed-Jamei
published in: Journal of Computational and Applied Mathematics
publication date: 2010-04-21T00:00:00Z
review text: A basic class of symmetric orthogonal functions with six parameters is introduced. It is obtained by using an extended theorem for Sturm-Liouville problems, and its standard properties are deduced. Indeed, it is shown that the new class satisfies the differential equation \[ x^2(p x^2+q) \varphi_n''(x)+ x(r x^2+s)\varphi_n'(x)-(a_nx^2+ (-1)^nc +d)\varphi_n(x)=0, \] in which \(c,d\) are two free parameters and \(-a_n\) denotes eigenvalues corresponding to the defined class. Four orthogonal sub-classes are introduced and a detailed study of their properties is given. These four sub-classes generalize the generalized ultraspherical polynomials, the generalized Hermite polynomials and two sequences of finite classical symmetric orthogonal polynomials with weight functions \(x^{-2a}(1+x^2)^{-b}\) and \(x^{-2a}\exp(\frac{-1}{x^2})\) on \((- \infty, \infty)\), respectively.
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5696582
zbMATH Keywords: extended Sturm-Liouville theorem for symmetric functions
zbMATH Keywords: symmetric orthogonal functions
zbMATH Keywords: Favard's theorem
zbMATH Keywords: generalized ultraspherical polynomials
zbMATH Keywords: generalized Hermite polynomials
zbMATH Keywords: two sequences of finite classical symmetric orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: j.cam.2009.12.025
OpenAlex ID: W1994516082
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DOI: 10.1016/J.CAM.2009.12.025
description: scientific article; zbMATH DE number 5696582