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 $L^{(\alpha)}_n(\nu z)$ as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane $\text{Re}\; z\geq \delta_1, 0<\delta_1<1$, which contains the critical point z=1; the other expansion holds uniformly in a left half-plane $\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1$, 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 $L^{(\alpha)}_n(\nu z)$.