Some monotonicity properties of the zeros of ultraspherical polynomials
type: Item
instance of: scholarly article
title: Some monotonicity properties of the zeros of ultraspherical polynomials
zbMATH Open document ID: 0611.33011
DOI: 10.1007/BF01949060
published in: Acta Mathematica Hungarica
publication date: 1986-01-01T00:00:00Z
review text: Using the Sturm comparison theorem, the authors prove results concerning the signs of the determinants \[ x_{ni}(\lambda)x_{m,j+\ell}(\lambda)- x_{mj}(\lambda)x_{n,i+\ell}(\lambda) \] and \[ \theta_{ni}(\lambda)\theta_{nk}(\lambda ')- \theta_{nk}(\lambda)\theta_{ni}(\lambda '), \] on the monotonicity in n of \(x_{nk}(\lambda)/x_{ni}(\lambda)\) and on the monotonicity in \(\ell\) of \(x_{n,i+\ell}(\lambda)/x_{n,j+\ell}(\lambda)\), where \(x_{n,k}(\lambda)\) is the k-th positive zero of the ultraspherical (or Gegenbauer) polynomial \(P_ n^{(\lambda)}(x)\), and \(\theta_{nk}(\lambda)=\arccos x_{nk}(\lambda)\).
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 3987687
zbMATH Keywords: Gegenbauer polynomial
zbMATH Keywords: ultraspherical polynomials
zbMATH Keywords: Sturm comparison theorem
zbMATH Keywords: monotonicity
author: Á. Elbert
author: A. Laforgia
reviewed by: Martin E. Muldoon
MaRDI profile type: Publication
cites work: On the Spacing of the Zeros of Some Classical Orthogonal Polynomials
cites work: A Monotonic Property for the Zeros of Ultraspherical Polynomials
full work available at URL: https://doi.org/10.1007/bf01949060
OpenAlex ID: W2007396565
description: scientific article; zbMATH DE number 3987687