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 as n →∞, where ν =4 n +2 α +2. One expansion holds uniformly in a right half-plane , which contains the critical point z =1; the other expansion holds uniformly in a left half-plane , 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 .