On the zeros of Jacobi polynomials
type: Item
instance of: scholarly article
title: On the zeros of Jacobi polynomials
zbMATH Open document ID: 0814.33006
DOI: 10.1007/BF01874695
author: Á. Elbert
author: A. Laforgia
author: Lucia G. Rodono
published in: Acta Mathematica Hungarica
publication date: 1994-09-13T00:00:00Z
review text: For \(\alpha>-1\), \(\beta>-1\), let \(x_{ni}= x_{ni} (\alpha, \beta)\) denote the \(i\)-th zero, in decreasing order, of the Jacobi polynomial \(P_ n^{(\alpha, \beta)} (x)\): \[ 1> x_{n1}> x_{n2}> \dots >x_{nn} >-1. \] Here the authors present a procedure based on the Sturm comparison theorem to obtain the following main result for \(x_{ni}\). For \(i=1,2, \dots,n\), let \(x_{ni}= x_{ni} (\alpha, \beta)\) be the \(i\)-th zero in decreasing order, of the Jacobi polynomial \(P_ n^{(\alpha, \beta)} (x)\). Then for \(\alpha >-{1\over 2}\), \(\beta> - {1\over 2}\) the following inequalities \[ \xi_ i< x_{ni}< \xi_{i-1}, \qquad i=1,2, \dots,n \] hold, where \(\xi_ 0, \xi_ 1,\dots, \xi_ n\) are the zeros of the function \[ (1- x^ 2)^{1/2} [(a-x) (x-b)]^{- 1/4} \sin \varphi(x) \] such that \[ \varphi(x)= (2n+ \overline {\alpha}+ \overline {\beta}) \arctan t- \overline {\alpha} \arctan \biggl( \sqrt {{{1-b} \over {1-a}} t} \biggr)- \overline {\beta} \arctan \biggl( \sqrt {{{1+b} \over {1+a}} t} \biggr) \] and \(\overline {\alpha}= \alpha+ {1\over 2}\) and \(\overline {\beta}= \beta+ {1\over 2}\).
reviewed by: Arun Kumar Agarwal
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 638210
zbMATH Keywords: Sturm comparison theorem
MaRDI profile type: Publication
cites work: Upper bounds for the zeros of ultraspherical polynomials
cites work: On the Zeros of Jacobi Polynomials P ( | alpha n , | beta n ) n (X)
full work available at URL: https://doi.org/10.1007/bf01874695
OpenAlex ID: W2056415549
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description: scientific article; zbMATH DE number 638210