Charlier polynomials

Short description: Orthogonal 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

$C_n(x;\mu )={}_2{F}_0(-n,-x;-;-1/\mu )=(-1{)}^{n}n!L_n^{(-1-x)}(-\frac{1}{\mu }),$

where $L$ are generalized Laguerre polynomials. They satisfy the following orthogonality relation in the Hilbert space of square summable sequences associated with the Poisson distribution with parameter $\mu$

$e^{\mu }⟨C_n(\cdot ,\mu ),C_m(\cdot ,\mu )⟩=\sum _{x=0}^{\mathrm{\infty }}\frac{{\mu }^x}{x!}{C}_n(x;\mu )C_m(x;\mu )=e^{\mu }{\mu }^{-n}n!{\delta }_{nm},\mu >0,$

where ${\delta }_{nm}$ is the Kronecker delta. They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion.

See also

• Wilson polynomials, a generalization of Charlier polynomials.

References

• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18.19 
• Szegő, Gabor (1939), Orthogonal Polynomials, Colloquium Publications – American Mathematical Society, ISBN 978-0-8218-1023-1