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On generalized Charlier polynomials. (English) Zbl 0537.33009

The Charlier polynomials are orthogonal polynomials of confluent hypergeometric type which arise in certain aspects of probability theory. Two generalizations of these polynomials are discussed here which may be written \[ C^{(a)}_{n,m}(x)=(-a)^ n_{m+1}F_ 0(-n/m,(1- n)/m,...,(m-n-1)/m,x;-;m^ m/a^ m) \] and \[ C_ n^{(a)}(m;x)=\sum^{[n/m]}_{k=0}\left( \begin{matrix} n\\ mk\end{matrix} \right)\left( \begin{matrix} x\\ n-mk\end{matrix} \right)((mk)!/k!)(-a)^ k. \] Although orthogonality relations relative to these two sets of polynomials are not given, interesting generating functions and pure recurrence relations are deduced all of which reduce to the corresponding formulae appertaining to the Charlier polynomials when m is put equal to unity.
Reviewer: H.Exton

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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