Charlier polynomials

In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by

${\displaystyle C_{n}(x;\mu )={}_{2}F_{0}(-n,-x;-;-1/\mu )=(-1)^{n}n!L_{n}^{(-1-x)}\left(-{\frac {1}{\mu }}\right),}${\displaystyle C_{n}(x;\mu )={}_{2}F_{0}(-n,-x;-;-1/\mu )=(-1)^{n}n!L_{n}^{(-1-x)}\left(-{\frac {1}{\mu }}\right),}

where ${\displaystyle L}${\displaystyle L} are generalized Laguerre polynomials. They satisfy the orthogonality relation

${\displaystyle \sum _{x=0}^{\infty }{\frac {\mu ^{x}}{x!}}C_{n}(x;\mu )C_{m}(x;\mu )=\mu ^{-n}e^{\mu }n!\delta _{nm},\quad \mu >0.}${\displaystyle \sum _{x=0}^{\infty }{\frac {\mu ^{x}}{x!}}C_{n}(x;\mu )C_{m}(x;\mu )=\mu ^{-n}e^{\mu }n!\delta _{nm},\quad \mu >0.}

They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion.

• Wilson polynomials, a generalization of Charlier polynomials.

• C. V. L. Charlier (1905–1906) Über die Darstellung willkürlicher Funktionen, Ark. Mat. Astr. och Fysic 2, 20.
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248 .
• Szegő, Gabor (1939), Orthogonal Polynomials, Colloquium Publications – American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517 {{citation}}: ISBN / Date incompatibility (help)

• Orthogonal polynomials
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