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

Abstract

We consider the uniform asymptotics of polynomials orthogonal on [0,∞) with respect to the exponential weight w(x)=x^αe^−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].