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 ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 ) n n ! L n ( − 1 − x ) ( − 1 μ ) , {\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 L {\displaystyle L} are generalized Laguerre polynomials. They satisfy the orthogonality relation

∑ x = 0 ∞ μ x x ! C n ( x ; μ ) C m ( x ; μ ) = μ − n e μ n ! δ n m , μ > 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.