Charlier polynomials

In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier in 1905.¹ They are given in terms of the generalized hypergeometric function by

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

where ${\displaystyle L}$ 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 ${\displaystyle \mu }$

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

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