GLOBAL ASYMPTOTICS OF THE HAHN POLYNOMIALS

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; α, β, N) as the degree n grows to infinity, when the parameters α and β are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann–Hilbert approach introduced by Deift and Zhou.