Charlier polynomials

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: