Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:153

Article; Proceedings Paper

Dimitrov, DK

关键词: ultraspherical polynomials;    Laguerre polynomials;    zeros;    convexity;    monotonicity;   

DOI  :  10.1016/S0377-0427(02)00645-3

来源: Elsevier

【 摘 要 】

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 Science B.V.

【 授权许可】

Free