# Charlier polynomials

Polynomials that are orthogonal on the system of non-negative integer points with an integral weight $d \sigma ( x)$, where $\sigma ( x)$ is a step function with jumps defined by the formula

$$j( x) = e ^ {-a} \frac{a ^ {x} }{x!} ,\ \ x = 0, 1 \dots \ \ a > 0.$$

The orthonormal Charlier polynomials have the following representations:

$$P _ {n} ( x; a) = \sqrt { \frac{a ^ {n} }{n!} } \sum _ { k= 0} ^ { n } (- 1) ^ {n-k} \left ( \begin{array}{c} n \\ k \end{array} \right ) k! a ^ {-k} \left ( \begin{array}{c} x \\ k \end{array} \right ) =$$

$$= \ a ^ {n / 2 } ( n!) ^ {- 1 / 2 } [ j( x)] ^ {-1} \Delta ^ {n} j ( x- n).$$

The Charlier polynomials are connected with the Laguerre polynomials by

$$P _ {n} ( x; a) = \sqrt {n! \over {a ^ {n} } } L _ {n} ^ {( x- n)} ( a) = \ \sqrt {n! \over {a ^ {n} } } L _ {n} ( a; x- n).$$

Introduced by C. Charlier . Since the function $j( x)$ defines a Poisson distribution, the polynomials $\{ P _ {n} ( x; a) \}$ are called Charlier–Poisson polynomials.

How to Cite This Entry:
Charlier polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Charlier_polynomials&oldid=51073
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article