Global asymptotics of krawtchouk polynomials - A riemann-hilbert approach

In this paper, we study the asymptotics of the Krawtchouk polynomials K_n^N (z;p,q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c (0, 1) as n → ∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou. © Springer-Verlag Berlin Heidelberg 2007.