Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:416

Article

Atia, M. J.¹  Martinez-Finkelshtein, A.^2,3  Martinez-Gonzalez, P.²  Thabet, F.⁴ 

[1] Fac Sci Gabes, Dept Math, Gabes, Tunisia

[2] Univ Almeria, Dept Math, Almeria, Spain

[3] Univ Granada, Inst Carlos I Fis Teor & Computac, E-18071 Granada, Spain

[4] ISSAT Gabes, Gabes, Tunisia

关键词: Trajectories and orthogonal trajectories of a quadratic differential;    Riemann-Hilbert problems;    Generalized Laguerre polynomials;    Strong and weak asymptotics;    Logarithmic potential;    Equilibrium;   

DOI  :  10.1016/j.jmaa.2014.02.040

来源: Elsevier

【 摘 要 】

In this paper we study the asymptotics (as n -> infinity) of the sequences of Laguerre polynomials with varying complex parameters a depending on the degree n. More precisely, we assume that alpha(n) = nA(n), and lim(n) A(n) = A is an element of C. This study has been carried out previously only for alpha(n) is an element of R, but complex values of A introduce an asymmetry that makes the problem more difficult. The main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality, which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane, which may have an independent interest. While the weak asymptotics is obtained by reduction to the theorem of Gonchar-Rakhmanov-Stahl, the strong asymptotic results are derived via the non-commutative steepest descent analysis based on the Riemann-Hilbert characterization of the Laguerre polynomials. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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