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

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

where $L$ are generalized Laguerre polynomials. They satisfy the orthogonality relation

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

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.
• .