Monotonicity of zeros of Jacobi polynomials
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Data
2007-11-01
Autores
Dimitrov, Dimitar K. [UNESP]
Rafaeli, Fernando R. [UNESP]
Título da Revista
ISSN da Revista
Título de Volume
Editor
Elsevier B.V.
Resumo
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 B.V. All rights reserved.
Descrição
Palavras-chave
zeros, Jacobi polynomials, monotonicity
Como citar
Journal of Approximation Theory. San Diego: Academic Press Inc. Elsevier B.V., v. 149, n. 1, p. 15-29, 2007.