Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

• Autor(a) principal: Dimitrov, D. K.
• Data de Publicação: 2003
• Tipo de documento: Conjunto de dados
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP (dados de pesquisa)
• Texto Completo: http://dx.doi.org/10.1016/S0377-0427(02)00645-3; http://hdl.handle.net/11449/21697
• topic: ultraspherical polynomials; Laguerre polynomials; zeros; convexity; monotonicity

Let C-n(lambda)(x), n = 0, 1,..., lambda > -1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal. in (-1, 1) with respect to the weight function (1 - x(2))(lambda-1/2). Denote by X-nk(lambda), k = 1,....,n, the zeros of C-n(lambda)(x) enumerated in decreasing order. In this short note, we prove that, for any n is an element of N, the product (lambda + 1)(3/2)x(n1)(lambda) is a convex function of lambda if lambda greater than or equal to 0. The result is applied to obtain some inequalities for the largest zeros of C-n(lambda)(x). If X-nk(alpha), k = 1,...,n, are the zeros of Laguerre polynomial L-n(alpha)(x), also enumerated in decreasing order, we prove that x(n1)(lambda)/(alpha + 1) is a convex function of alpha for alpha > - 1. (C) 2002 Published by Elsevier B.V. B.V.