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Uniform asymptotic expansions for Meixner polynomials. (English) Zbl 0906.41020
Meixner polynomials ${m}_{n}\left(x;\beta ,c\right)$ are considered for large values of $n$. Two uniform expansions of ${m}_{n}\left(n\alpha ;\beta ,c\right)$ are given, in terms of parabolic cylinder functions, one holding uniformly for $\alpha \in \left[\epsilon ,1+a\right]$ and the other one for $\alpha \in \left[1-b,M\right]$, where $\epsilon ,a$ and $b$ are small positive numbers and $M<\infty$. The results are obtained by steepest descent methods and include five asymptotic formulas given earlier by W. M. Y. Goh.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C45 Orthogonal polynomials and functions of hypergeometric type