# Charlier polynomials

From Encyclopedia of Mathematics

Polynomials that are orthogonal on the system of non-negative integer points with an integral weight , where is a step function with jumps defined by the formula

The orthonormal Charlier polynomials have the following representations:

The Charlier polynomials are connected with the Laguerre polynomials by

Introduced by C. Charlier [1]. Since the function defines a Poisson distribution, the polynomials are called Charlier–Poisson polynomials.

#### References

[1] | C. Charlier, "Application de la théorie des probabilités à l'astronomie" , Paris (1931) |

[2] | H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) |

[3] | G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) |

#### Comments

In the formula above, denotes taking first differences, i.e. . Another common notation and an expression by hypergeometric functions is:

**How to Cite This Entry:**

Charlier polynomials.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Charlier_polynomials&oldid=13248

This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article