Abstract
Let \(K_{n}^{N}(x;p,q)\) be the Krawtchouk polynomials and μ = 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 μ in [1,∞), and is given in terms of the confluent hypergeometric functions. Asymptotic approximations are also obtained for the zeros of \(K_{n}^{N}(x;p,q)\) in various cases depending on the values of p, q and μ.
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The research of the first author is partially supported by Liu Bie Ju Center for Mathematics Sciences and by Chinese NNSF grant No. 10271031. The research of the second author is partially supported by grants from the Research Grant Council of Hong Kong.
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Qiu, WY., Wong, R. Asymptotic Expansion of the Krawtchouk Polynomials and their Zeros. Comput. Methods Funct. Theory 4, 189–226 (2004). https://doi.org/10.1007/BF03321065
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DOI: https://doi.org/10.1007/BF03321065