Monotonicity of zeros of Jacobi polynomials

JOURNAL OF APPROXIMATION THEORY

卷:149

Article

Dimitrov, Dimitar K.¹  Rafaeli, Fernando R.¹ 

[1] Univ Estadual Paulista, Dept Ciencias Computacao & Estat, IBILCE, Sao Paulo, Brazil

关键词: zeros;    Jacobi polynomials;    monotonicity;   

DOI  :  10.1016/j.jat.2007.04.004

来源: Elsevier

【 摘 要 】

Denote by x(nk)(alpha, beta), k = 1...., n, the zeros of the Jacobi polynornial P-n((alpha,beta)) (x). It is well known that x(nk)(alpha, beta) are increasing functions of beta and decreasing functions of alpha. In this paper we investigate the question of how fast the functions 1 - x(nk)(alpha, beta) decrease as beta increases. We prove that the products t(nk)(alpha, beta) := f(n)(alpha, beta) (1 - x(nk)(alpha, beta), where f(n)(alpha, beta) = 2n(2) + 2n(alpha + beta + 1) + (alpha + 1)(beta + 1) are already increasing functions of beta and that, for any fixed alpha > - 1, f(n)(alpha, beta) is the asymptotically extremal, with respect to n, function of beta that forces the products t(nk)(alpha, beta) to increase. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free