Global Asymptotics of Krawtchouk Polynomials – a Riemann-Hilbert Approach*

Abstract

In this paper, we study the asymptotics of the Krawtchouk polynomials $K^{N}_{n} {\left( {z;p,q} \right)}$ as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters $\frac{n}{N}$ 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.

Keywords

Global asymptotics / Krawtchouk polynomials / Parabolic cylinder functions / Airy functions / Riemann-Hilbert problems / 41A60 / 33C45

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