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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=\sum_{n=0}^\infty 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 $\mu=N/n \in [1,\infty)$. 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 $\mu$.

33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
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