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Charlier Polynomial

The orthogonal polynomials defined by

c_n^((mu))(x)=_2F_0(-n,-x;;-mu^(-1))
(1)
=((-1)^n)/(mu^n)(x-n+1)_n_1F_1(-n;x-n+1;mu),
(2)

where (x)_n is the Pochhammer symbol (Koekoek and Swarttouw 1998). The first few are given by

c_0^((mu))(x)=1
(3)
c_1^((mu))(x)=1-x/mu
(4)
c_2^((mu))(x)=(x^2+mu^2-x(1+2mu))/(mu^2).
(5)

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References

Koekoek, R. and Swarttouw, R. F. "Charlier." §1.12 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 49-50, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 115, 1998.

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Charlier Polynomial

Cite this as:

Weisstein, Eric W. "Charlier Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharlierPolynomial.html

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