Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 4 (2004), No. 1, 189--226 Copyright Heldermann Verlag 2004 Asymptotic Expansion of the Krawtchouk Polynomials and their Zeros Wei-Yuan Qiu wyqiu@fudan.edu.cn , Department of Mathematics, Fudan University, Shanghai, People's Republic of China. Roderick Wong mawong@cityu.edu.hk , Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong. [Abstract-pdf] [Abstract-ps] Let $K_n^N(x;p,q)$ be the Krawtchouk polynomials and $\mu=N/n$. An asymptotic expansion is derived for $K_n^N(x;p,q)$, when $x$ is a fixed number. This expansion holds uniformly for $\mu$ in $[1,\infty)$, and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of $K_n^N(x;p,q)$ in various cases depending on the values of $p,q$ and $\mu$. Keywords: Krawtchouk polynomials, asymptotic expansions, confluent hypergeometric functions, zeros. MSC 2000: 33C45, 41A60. [FullText-pdf (544 K)] [FullText-ps (588 K)]