Charlier polynomials explained

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} ^{n} ^{0 ( - n, - x; - ; -1 /\mu) = ( -1 )} n! L ^{( -1- x)} _{n} \left( - (1)/(\mu) \right), where L are generalized Laguerre polynomials. They satisfy the orthogonality relation\sum ^{inf ty} _{x =0} (\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.