Asymptotic Expansion of the Krawtchouk Polynomials and their Zeros

Let K N n (x;p,q) KnN(x;p,q) be the Krawtchouk polynomials and μ = N/n. An asymptotic expansion is derived for K N n (x;p,q) KnN(x;p,q) , when x is a fixed number. This expansion holds uniformly for μ in [1,∞), and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of K N n (x;p,q) KnN(x;p,q) in various cases depending on the values of p, q and μ.