Computational Methods and Function Theory 4 (2004), No. 1, 189--226
Copyright Heldermann Verlag 2004
Asymptotic Expansion of the Krawtchouk Polynomials and their Zeros
email@example.com , Department of Mathematics, Fudan University, Shanghai, People's Republic of China.
firstname.lastname@example.org , Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong.
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.
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