Monotonicity properties of the zeros of ultraspherical polynomials
type: Item
instance of: scholarly article
title: Monotonicity properties of the zeros of ultraspherical polynomials
zbMATH Open document ID: 0923.33004
author: Á. Elbert
author: Panayiotis D. Siafarikas
published in: Journal of Approximation Theory
publication date: 1999-10-28T00:00:00Z
review text: Let \(x_{n,k}^{(\lambda)}\), \(k=1,2,\dots,[n,2]\), denote the \(k\)-th positive zero in increasing order of the ultraspherical polynomial \(P_n^{(\lambda)}(x)\). The authors prove that the function \([\lambda+(2n^2+1)/(4n+2)]^{1/2}x_{n,k}^{(\lambda)}\) is increasing for \(\lambda>-1/2\). This proves the Ismail-Letessier-Askey conjecture [\textit{M. E. H. Ismail}, in \(q\)-series and partitions, Proc. Workshop, Minneapolis/MN(USA) 1988, IMA Vol. MAth. Appl. 18, 177-190 (1989; Zbl 0698.33008)], the function \((\lambda+1)^{1/2}x_{n,k}(\lambda)\) increases as \(\lambda\) increases for \(\lambda>-1/2\).
Mathematics Subject Classification ID: 33C55
zbMATH DE Number: 1286386
zbMATH Keywords: ultraspherical polynomial
reviewed by: Daniel Duverney
MaRDI profile type: Publication
full work available at URL: jath.1996.3259
OpenAlex ID: W2002764591
cites work: Inequalities and Numerical Bounds for Zeros of Ultraspherical Polynomials
cites work: On a conjecture concerning monotonocity of zeros of ultraspherical polynomials
cites work: A Note to the Paper of Ahmed, Muldoon and Spigler
cites work: Upper bounds for the zeros of ultraspherical polynomials
cites work: Asymptotic Formulas for Ultraspherical Polynomials P (λ) n (X) and their Zeros for Large Values of λ
cites work: On the Derivative with Respect to a Parameter of a Zero of a Sturm–Liouville Function
cites work: Q5827353
cites work: Q3671491
cites work: Q5330830
cites work: Q3475560
cites work: A Monotonic Property for the Zeros of Ultraspherical Polynomials
cites work: Q3705717
DOI: 10.1006/JATH.1996.3259
description: scientific article; zbMATH DE number 1286386