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Polynomials that are orthogonal on the system of non-negative integer points with an integral weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217902.png" /> is a step function with jumps defined by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217903.png" /></td> </tr></table>
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Polynomials that are orthogonal on the system of non-negative integer points with an integral weight  $  d \sigma ( x) $,
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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:
 
The orthonormal Charlier polynomials have the following representations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217904.png" /></td> </tr></table>
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$$
 +
P _ {n} ( x; a)  = \sqrt {
 +
\frac{a  ^ {n} }{n!}
 +
} \sum _ { k= } 0 ^ { n }  (- 1)  ^ {n-} k
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) k! a  ^ {-} k \left ( \begin{array}{c}
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x \\
 +
k
 +
\end{array}
 +
\right ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217905.png" /></td> </tr></table>
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$$
 +
= \
 +
a ^ {n / 2 } ( n!) ^ {- 1 / 2 } [ j( x)]  ^ {-} 1 \Delta  ^ {n} j ( x- n).
 +
$$
  
 
The Charlier polynomials are connected with the [[Laguerre polynomials|Laguerre polynomials]] by
 
The Charlier polynomials are connected with the [[Laguerre polynomials|Laguerre polynomials]] by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217906.png" /></td> </tr></table>
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$$
 +
P _ {n} ( x; a)  = \sqrt {n! over {a  ^ {n} } } L _ {n}  ^ {(} x- n) ( a)  = \
 +
\sqrt {n! over {a  ^ {n} } } L _ {n} ( a; x- n).
 +
$$
  
Introduced by C. Charlier [[#References|[1]]]. Since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217907.png" /> defines a Poisson distribution, the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217908.png" /> are called Charlier–Poisson polynomials.
+
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====
 
====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,  "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>
  
 +
====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 )  = \
  
====Comments====
+
\frac{P _ {n} ( x ;  a ) }{P _ {n} ( 0 ;  a ) }
In the formula above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c0217909.png" /> denotes taking first differences, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c02179010.png" />. Another common notation and an expression by hypergeometric functions is:
+
  = \
 
+
{} _ {2} F _ {0} ( - n , - x ; - a  ^ {-} 1 ) .
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021790/c02179011.png" /></td> </tr></table>
+
$$

Revision as of 16:43, 4 June 2020


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, "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, $ \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