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In [[mathematics]], '''Charlier polynomials''' (also called '''Poisson–Charlier polynomials''') are a family of [[orthogonal polynomials]] introduced by [[Carl Charlier]].
They are given in terms of the [[generalized hypergeometric function]] by
:<math>C_n(x; \mu)= {}_2F_0(-n,-x,-1/\mu)=(-1)^n n! L_n^{(-1-x)}\left(-\frac 1 \mu \right),\,</math>
where <math>L</math> are [[Laguerre polynomials]]. They satisfy the [[orthogonality relation]]
:<math>\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0.</math>


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== See also ==
* [[Wilson polynomials]], a generalization of Charlier polynomials.
 
== References ==
* C. V. L. Charlier (1905–1906) ''Über die Darstellung willkürlicher Funktionen'', Ark. Mat. Astr. och Fysic 2, 20.
* {{dlmf|id=18.19|title=Hahn Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
* {{Citation | authorlink=Gábor Szegő | last1=Szegő | first1=Gabor | title=Orthogonal Polynomials | publisher=Colloquium Publications – American Mathematical Society | isbn=978-0-8218-1023-1 | mr=0372517 | year=1939}}
 
[[Category:Orthogonal polynomials]]
 
 
{{Algebra-stub}}

Revision as of 18:18, 19 March 2013

In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by

where are Laguerre polynomials. They satisfy the orthogonality relation

See also

References

  • C. V. L. Charlier (1905–1906) Über die Darstellung willkürlicher Funktionen, Ark. Mat. Astr. och Fysic 2, 20.
  • Template:Dlmf
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