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

References

[1]	C. Charlier, "Applications de la théorie des probabilités à l'astronomie" , Paris (1931) Zbl 57.0620.03
[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, $ \Delta $ denotes taking first differences, i.e. $ \Delta f ( x) = f ( x + 1 ) - f ( x) $. Another common notation and an expression by hypergeometric functions is:

$$ C _ {n} ( x ; a ) = \ \frac{P _ {n} ( x ; a ) }{P _ {n} ( 0 ; a ) } = \ {} _ {2} F _ {0} ( - n , - x ; - a ^ {-1} ) . $$