A uniform asymptotic expansion for Krawtchouk polynomials

We study the asymptotic behavior of the Krawtchouk polynomial K^(N)_n(x; p, q) as n→∞. With x≡λN and ν=n/N, an infinite asymptotic expansion is derived, which holds uniformly for λ and ν in compact subintervals of (0, 1). This expansion involves the parabolic cylinder function and its derivative. When ν is a fixed number, our result includes the various asymptotic approximations recently given by M. E. H. Ismail and P. Simeonov. © 2000 Academic Press.