Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials
type: Item
instance of: scholarly article
title: Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials
zbMATH Open document ID: 1017.33004
DOI: 10.1016/S0377-0427(02)00645-3
author: Dimitar K. Dimitrov
published in: Journal of Computational and Applied Mathematics
publication date: 2003-05-22T00:00:00Z
review text: In this paper it is proved that for any \(n\in\mathbb{N}\), the product \((\lambda+1)^{3/2} x_{n1}(\lambda)\) is a convex function of \(\lambda\) if \(\lambda\geq 0\), where \(x_{nk}(\lambda)\), \(k= 1,\dots, n\) are denoting zeros of ultraspherical polynomials \(C^\lambda_n(x)\) enumerated in decreasing order. This result is applied to obtain some inequalities for the largest zeros of \(C^\lambda_n(x)\). If \(x_{nk}(\alpha)\), \(k= 1,\dots, n\) are zeros of Laguerre polynomial \(L^\alpha_n(x)\) (also enumerated in decreasing order) it is proved that \(x_{n1}(\lambda)/(\alpha+ 1)\) is a convex function of \(\alpha\) for \(\alpha> -1\).
reviewed by: Som Prakash Goyal
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 1915231
zbMATH Keywords: ultraspherical polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: zeros
zbMATH Keywords: convexity
zbMATH Keywords: monotonicity
MaRDI profile type: Publication
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full work available at URL: s0377-0427(02)00645-3
OpenAlex ID: W1978614574
description: scientific article; zbMATH DE number 1915231