View source for Charlier polynomials

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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  C. Charlier, "Applications de la théorie des probabilités à l'astronomie" , Paris  (1931) {{ZBL|57.0620.03}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
</table>

====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} ) .
$$