Global asymptotics of orthogonal polynomials associated with [formula omitted

Abstract

In this paper, we consider the asymptotics of polynomials orthogonal with respect to the weight function w ( x ) = | x | 2 α e − Q ( x ) , α > − 1 2 , where Q ( x ) = ∑ k = 0 2 m q k x k , q 2 m > 0 , m > 0 is a polynomial of degree 2 m . Globally uniform asymptotic expansions are obtained for z in four regions. These regions together cover the whole complex z -plane. Due to the singularity of | x | 2 α , the expansion in the region containing the origin involves Bessel functions. We also study the asymptotic behavior of the leading coefficients and the recurrence coefficients of these polynomials. Our approach is based on a modified version of the steepest descent method for Riemann–Hilbert problems introduced by Deift and Zhou [P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Asymptotics for the mKdV equation, Ann. of Math. 137 (1993) 295–368].