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Uniform asymptotic expansion of Charlier polynomials. (English) Zbl 0846.41025

Summary: The Charlier polynomials \(C_n^{(a)} (x)\) form an orthogonal system on the positive real line \(x>0\) with respect to the distribution \(d\alpha (x)\), where \(\alpha (x)\) is a step function with jumps at the non-negative integers. Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. An infinite asymptotic expansion is derived for \(C_n^{(a)} (n\beta)\), as \(n\to \infty\), which holds uniformly for \(0< \varepsilon\leq \beta\leq M< \infty\). Our result includes as special cases all seven asymptotic formuals recently given by W. M. Y. Goh.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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