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Asymptotic expansion of the Krawtchouk polynomials and their zeros. (English) Zbl 1053.33005
The generating function ${\left(1-pw\right)}^{N-x}{\left(1+qw\right)}^{x}={\sum }_{n=0}^{\infty }{K}_{n}^{N}\left(x;p,q\right){w}^{n}$ and Cauchy’s formula are used for obtaining an asymptotic expansion for large $n$ for the Krawtchouk polynomials ${K}_{n}^{N}\left(x;p,q\right)$. The expansion holds for fixed or bounded $x$ and is uniformly for $\mu =N/n\in \left[1,\infty \right)$. The main approximants are confluent hypergeometric functions. Asymptotic approximations are also derived for the zeros of ${K}_{n}^{N}\left(x;p,q\right)$ for various cases depending on the values of $p$, $q$, and $\mu$.
MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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