On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

JOURNAL OF APPROXIMATION THEORY

卷:162

Article

Moreno, A. Foulquie²  Martinez-Finkelshtein, A.^3,4  Sousa, V. L.¹ 

[1] Univ Aveiro, AE Escariz & Res Unity Matemat & Aplicacoes, P-3810193 Aveiro, Portugal

[2] Univ Aveiro, Dept Math & Res Unity Matemat & Aplicacoes, P-3810193 Aveiro, Portugal

[3] Univ Almeria, Dept Stat & Appl Math, Almeria, Spain

[4] Univ Granada, Ist Carlos Fis Teor & Computac 1, E-18071 Granada, Spain

关键词: Orthogonal polynomials;    Asymptotics;    Riemann-Hilbert method;    Steepest descent;    Recurrence coefficients;    Generalized Jacobi weights;   

DOI  :  10.1016/j.jat.2009.08.006

来源: Elsevier

【 摘 要 】

In 1995, Magnus [15] posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1, 1] of the form [GRAPHICS] with A, B > 0, alpha,beta,gamma > -1, and x(0) is an element of (-1, 1). We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1, 11 and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at x(0) has to be carried out in terms of confluent hypergeometric functions. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free