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$$  
 
$$  
j( x)  =  e  ^ {-} a
+
j( x)  =  e  ^ {-a}  
 
\frac{a  ^ {x} }{x!}
 
\frac{a  ^ {x} }{x!}
 
  ,\ \  
 
  ,\ \  
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P _ {n} ( x;  a)  =  \sqrt {
 
P _ {n} ( x;  a)  =  \sqrt {
 
\frac{a  ^ {n} }{n!}
 
\frac{a  ^ {n} }{n!}
  } \sum _ { k= } 0 ^ { n }  (- 1)  ^ {n-} k
+
  } \sum _ { k= 0} ^ { n }  (- 1)  ^ {n-k}
 
\left ( \begin{array}{c}
 
\left ( \begin{array}{c}
 
n \\
 
n \\
 
  k  
 
  k  
 
\end{array}
 
\end{array}
  \right ) k! a  ^ {-} k \left ( \begin{array}{c}
+
  \right ) k! a  ^ {-k} \left ( \begin{array}{c}
 
x \\
 
x \\
 
  k  
 
  k  
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$$  
 
$$  
 
= \  
 
= \  
a ^ {n / 2 } ( n!) ^ {- 1 / 2 } [ j( x)]  ^ {-} 1 \Delta  ^ {n} j ( x- n).
+
a ^ {n / 2 } ( n!) ^ {- 1 / 2 } [ j( x)]  ^ {-1} \Delta  ^ {n} j ( x- n).
 
$$
 
$$
  
The Charlier polynomials are connected with the [[Laguerre polynomials|Laguerre polynomials]] by
+
The Charlier polynomials are connected with the [[Laguerre polynomials]] by
  
 
$$  
 
$$  
P _ {n} ( x;  a)  =  \sqrt {n! over {a  ^ {n} } } L _ {n}  ^ {(} x- n) ( a)  = \  
+
P _ {n} ( x;  a)  =  \sqrt {n! \over {a  ^ {n} } } L _ {n}  ^ {( x- n)} ( a)  = \  
\sqrt {n! over {a  ^ {n} } } L _ {n} ( a;  x- n).
+
\sqrt {n! \over {a  ^ {n} } } L _ {n} ( a;  x- n).
 
$$
 
$$
  
 
Introduced by C. Charlier [[#References|[1]]]. Since the function  $  j( x) $
 
Introduced by C. Charlier [[#References|[1]]]. Since the function  $  j( x) $
defines a Poisson distribution, the polynomials  $  \{ P _ {n} ( x;  a) \} $
+
defines a [[Poisson distribution]], the polynomials  $  \{ P _ {n} ( x;  a) \} $
 
are called Charlier–Poisson polynomials.
 
are called Charlier–Poisson polynomials.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Charlier,   "Application de la théorie des probabilités à l'astronomie" , Paris  (1931)</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>
+
<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====
 
====Comments====
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\frac{P _ {n} ( x ;  a ) }{P _ {n} ( 0 ;  a ) }
 
\frac{P _ {n} ( x ;  a ) }{P _ {n} ( 0 ;  a ) }
 
   = \  
 
   = \  
{} _ {2} F _ {0} ( - n , - x ;  - a  ^ {-} 1 ) .
+
{} _ {2} F _ {0} ( - n , - x ;  - a  ^ {-1} ) .
 
$$
 
$$

Latest revision as of 20:19, 16 March 2024


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:

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

How to Cite This Entry:
Charlier polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Charlier_polynomials&oldid=46325
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article