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Uniform asymptotic expansions for Charlier polynomials. (English) Zbl 0991.41020
The author investigates the asymptotic behavior of the Charlier polynomials $C_n^{(a)}$ as $n \to \infty$. The polynomials $C_n^{(a)}$ are not regarded as functions of $x$ with $a$ as a parameter, but rather with the roles reversed via a second-order linear differential equation in which $a$ is the (real or complex valued) independent variable and $x$ is a parameter.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C45 Orthogonal polynomials and functions of hypergeometric type 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
##### Keywords:
asymptotic expansion; Charlier polynomials
Full Text:
##### References:
 [1] Boyd, W. G. C.; Dunster, T. M.: Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. anal. 17, 422-450 (1986) · Zbl 0591.34048 [2] Chihara, T. S.: An introduction to orthogonal polynomials. (1978) · Zbl 0389.33008 [3] Goh, W. M. Y.: Plancherel--rotach asymptotics for the Charlier polynomials. Constr. approx. 14, 151-168 (1998) · Zbl 0894.33004 [4] Maejima, M.; Van Assche, W.: Probabilistic proofs of asymptotic formulas for some classical polynomials. Math. proc. Cambridge philos. Soc. 97, 499-510 (1985) · Zbl 0557.33006 [5] Olver, F. W. J.: Asymptotics and special functions. (1974) · Zbl 0303.41035 [6] Rui, B.; Wong, R.: Uniform asymptotic expansion of Charlier polynomials. Methods appl. Anal. 1, 109-134 (1994) · Zbl 0846.41025 [7] Szego?, G.: Orthogonal polynomials. Colloquim publications 23 (1975)