Global asymptotics of the Meixner polynomials

Using the steepest descent method for oscillatory Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math. 137 (1993), 295–368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals [Constr. Approx. 14 (1998), 113–150].