Perturbed zeros of classical orthogonal polynomials
type: Item
instance of: scholarly article
title: Perturbed zeros of classical orthogonal polynomials
zbMATH Open document ID: 0958.33005
author: N. Delaunay
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2001-04-10T00:00:00Z
review text: It is known that the set of \(N\) zeros of polynomials satisfying linear homogeneous second order differential equations is the solution of certain system of \(N\) nonlinear equations depending on a function related to one of the coefficients of the differential equation and, in the case of the classical orthogonal polynomials to the corresponding weight function. By perturbing such a function the author generates sets of perturbed zeros. This procedure yields a linear problem for the differences between the original and perturbed zeros. The matrix of this linear problem is the one used by Stieltjes to show the monotonic variation of the zeros of the classical orthogonal polynomials. The elementary method presented in this paper can be used to give a bound for the error produced in approximating the original set of zeros by the perturbed one. As simple examples the author obtains a bound for the euclidean norm of the vector of differences between the zeros of Hermite polynomials and those of (approximately scaled) Gegenbauer and factorial polynomials.
reviewed by: Ismail Taqi Ali
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 1501463
zbMATH Keywords: classical orthogonal polynomials
zbMATH Keywords: zeros
zbMATH Keywords: Stieltjes matrix
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1501463