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
 * $$C_n(x; \mu)= {}_2F_0(-n,-x;-;-1/\mu)=(-1)^n n! L_n^{(-1-x)}\left(-\frac 1 \mu \right),$$

where $$L$$ are generalized Laguerre polynomials. They satisfy the orthogonality relation
 * $$\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0.$$

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