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Asymptotic expansion of the Krawtchouk polynomials and their zeros. (English) Zbl 1053.33005
The generating function (1-pw) N-x (1+qw) x = n=0 K n N (x;p,q)w n and Cauchy’s formula are used for obtaining an asymptotic expansion for large n for the Krawtchouk polynomials K n N (x;p,q). The expansion holds for fixed or bounded x and is uniformly for μ=N/n[1,). The main approximants are confluent hypergeometric functions. Asymptotic approximations are also derived for the zeros of K n N (x;p,q) for various cases depending on the values of p, q, and μ.
MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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