Zeros of the Wronskian of Chebyshev and ultraspherical polynomials
type: Item
instance of: scholarly article
title: Zeros of the Wronskian of Chebyshev and ultraspherical polynomials
zbMATH Open document ID: 0779.30004
author: Karl Dilcher
author: Kenneth B. Stolarsky
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-01-18T00:00:00Z
review text: If \(P(z)\) is a polynomial then the Wronskian of \(P\) is defined to be \(WP(z)=P(z)P''(z)-(P'(z))^ 2\). In an earlier paper [J. Math. Anal. Appl. 162, No. 2, 430-451 (1991; Zbl 0748.30007)], the authors showed that if all zeros of \(P(z)\) lie on the interval \([-1,1]\) then all the zeros of \(WP(z)\) lie inside or on the unit circle. Here they study the Wronskian of the Chebyshev polynomials and more generally the ultraspherical polynomials of order \(0<\lambda\leq 1\). For \(P(z)=T_ n(z)\), they show that the zeros of \(WT_ n(z)\) lie inside the ellipse \((x/B_ n)^ 2+(y/A_ n)^ 2=1\), where \(A_ n\sim\log n/n\) and \(B_ n=(1+A^ 2_ n)^{1/2}\), and hence are very close to the real axis for large \(n\). They obtain a similar but somewhat less sharp result in the case \(P(z)=C^ \lambda_ n(z)\).
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 218632
zbMATH Keywords: Wronskian
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: ultraspherical polynomials
reviewed by: David W. Boyd
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072610
OpenAlex ID: W1968237685
cites work: Q5558293
cites work: Zeros of the Wronskian of a polynomial
cites work: On certain determinants whose elements are orthogonal polynomials
cites work: Q5546070
cites work: Some Inequalities Concerning Jacobi Polynomials
DOI: 10.1216/RMJM/1181072610
description: scientific article; zbMATH DE number 218632