On the roots of orthogonal polynomials and Euler-Frobenius polynomials
type: Item
instance of: scholarly article
title: On the roots of orthogonal polynomials and Euler-Frobenius polynomials
zbMATH Open document ID: 0855.42017
author: F. Dubeau
author: Jean Savoie
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-01-22T00:00:00Z
review text: Consider a family of polynomials recursively defined by \(P_0 (x) = x^l\) \((l\) any nonnegative integer) and \[ c_{n + 1} P_{n + 1} (x) = - 2r_n xP_n (x) + (1 - x^2) P_n' (x), \quad n \geq 0, \] where \(r_n > 0\) and \(c_n \in \mathbb{R}\backslash\{0\}\). For example, ultraspherical and Euler-Frobenius polynomials verify this scheme. The authors show that the roots of \(P_n\) are real, simple and belong to \((- 1,1)\). Interlacing properties of the zeros of \(P_n\) and \(P_{n + 1}\) (and \(P_n\) and \(P_{n + 2})\) are established. The method of proof is extended to the study of Hermite like polynomials given by \(H_0 (x) = x^l\) and \[ c_{n + 1} H_{n + 1} (x) = - 2xH_n (x) + H_n'(x), \] where \(c_n \in \mathbb{R}\backslash\{0\}\).
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 861806
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: roots
zbMATH Keywords: interlacing property
zbMATH Keywords: recurrence relations
zbMATH Keywords: Euler-Frobenius polynomials
zbMATH Keywords: Hermite like polynomials
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1399
OpenAlex ID: W2091986647
DOI: 10.1006/JMAA.1995.1399
description: scientific article; zbMATH DE number 861806