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Charlier polynomials

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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

:C_n(x; \mu)= {}_2F_0(-n,-x;-;-1/\mu)=(-1)^n n! L_n^{(-1-x)}\left(-\frac 1 \mu \right),

where L are generalized Laguerre polynomials. They satisfy the orthogonality relation

:\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.

They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion.

See also

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

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