Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

Abstract

In this paper, we study the asymptotic behavior of the Laguerre polynomials \(L_{n}^{(\alpha_{n})}(nz)\) as n→∞. Here αn is a sequence of negative numbers and −αn/n tends to a limit A>1 as n→∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane ℂ+={z:Im z≥0}. A corresponding expansion is also given for the lower half plane ℂ−={z:Im z≤0}. The two expansions hold, in particular, in regions containing the curve Γ in the complex plane, on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.