Global asymptotic expansions of the Laguerre polynomials - A Riemann-Hilbert approach

By using the steepest descent method for Riemann-Hilbert problems introduced by Deift-Zhou (Ann Math 137:295-370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial L_n ^(α)(νz) as n → ∞, where ν = 4n + 2α + 2. One expansion holds uniformly in a right half-plane Re z ≥ δ₁, 0 <δ₁ <1, which contains the critical point z = 1; the other expansion holds uniformly in a left half-plane Re z ≤ 1 - δ₂, 0 <δ₂ <1 - δ₁, which contains the other critical point z = 0. The two half-planes together cover the entire complex z-plane. The critical points z = 1 and z = 0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by L_n ^(α)(νz) . © 2008 Springer Science+Business Media, LLC.