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Charlier polynomials

Orthogonal polynomials

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated June 3, 2026

In mathematics, Charlier polynomials (also called Poisson-Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier in 1905. 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)}\left(-{\frac {1}{\mu }}\right),

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

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This article is adapted from the Wikipedia article “Charlier polynomials”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Wilson polynomials

In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James Wilson that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by p n = ( a + b ) n ( a + c ) n ( a + d ) n 4 F 3 ( − n a + b + c + d + n − 1 a − t a + t a + b a + c a + d ; 1 ) .