Asymptotic properties of Krawtchouk polynomials (Q1120710)

By utilizing a discrete analogue of Leibniz's theorem, a new uniform asymptotic form of the Krawtchouk polynomial \(K_ n(x;N,p)\) is established as \(n=O(N^{1/3})\to \infty\). Certain properties of the Hermite polynomial are also employed. Three corollaries are then deduced, the first giving extrema bounds of the Hermite function and the two others useful asymptotic results on the zeros of the Krawtchouk polynomial. This interesting paper is concluded by indicating an application to coding theory.