On a conjectured inequality for the largest zero of Jacobi polynomials
type: Item
instance of: scholarly article
title: On a conjectured inequality for the largest zero of Jacobi polynomials
zbMATH Open document ID: 1167.33003
author: Walter Gautschi
published in: Numerical Algorithms
publication date: 2009-04-29T00:00:00Z
review text: Recently [\textit{W. Gautschi} and \textit{P. Leopardi}, Numer. Algorithms 45, No.~1--4, 217--230 (2007; Zbl 1124.33009)], it was conjectured that, given \(\alpha>-1\), \(\beta>-1\), the largest zeros \(x_n^{(\alpha,\beta)}=\cos\theta_n^{(\alpha,\beta)}\), \(n=1,2,\dots\), of the Jacobi polynomials \(P_n^{(\alpha,\beta)}(x)\) satisfy the inequality \(n\theta_n^{(\alpha,\beta)}<(n+1)\theta_{n+1}^{(\alpha,\beta)}\) for all \(n>1\) iff this inequality holds true for \(n=1\). In the present paper, the author shows that the domain in the \((\alpha,\beta)\) plane of validity of the above inequality for all \(n\geq1\) requires some more detailed description. Namely, it is known that the \(n\)-large asymptotics of the Jacobi polynomial \(P_n^{(\alpha,\beta)}(x)\) in some region of the independent variable \(x\) can be described in terms of the Bessel function \(J_{\alpha}\). Taking into account four initial terms in the asymptotic expansion in \(n^{-1}\) of the ratio \(\theta_n^{(\alpha,\beta)}/\theta_{n+1}^{(\alpha,\beta)}\), the author observes a certain domain in the \((\alpha,\beta)\)-plane where the considered inequality is false for all sufficiently large \(n\) even though it holds true for \(n=1\). The paper is concluded by a new conjecture concerning the above inequality.
reviewed by: Andrei A. Kapaev
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5549491
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: zeros
zbMATH Keywords: asymptotics
zbMATH Keywords: inequalities
MaRDI profile type: Publication
full work available at URL: s11075-008-9207-6
OpenAlex ID: W2124574067
cites work: Luigi Gatteschi's work on asymptotics of special functions and their zeros
cites work: Conjectured inequalities for Jacobi polynomials and their largest zeros
DOI: 10.1007/S11075-008-9207-6
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description: scientific article; zbMATH DE number 5549491