Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

Article type: Research Article

Authors: Dai, Dan^; | Qiu, Weiyuan | Wang, Jun

Affiliations: Department of Mathematics, City University of Hong Kong, Hong Kong, China. E-mail: dandai@cityu.edu.hk | School of Mathematical Sciences, Fudan University, Shanghai, China. E-mails: {majwang, wyqiu}@fudan.edu.cn

Note: [] Corresponding author. E-mail: dandai@cityu.edu.hk

Abstract: We consider the uniform asymptotics of polynomials orthogonal on [0,∞) with respect to the exponential weight w(x)=xαe−Q(x), where α>−1 and Q(x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math. 137 (1993), 295–368].

Keywords: orthogonal polynomial, uniform asymptotic expansion, Laguerre polynomial, Riemann–Hilbert problem

DOI: 10.3233/ASY-141219

Journal: Asymptotic Analysis, vol. 89, no. 1-2, pp. 1-19, 2014