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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

where are generalized Laguerre polynomials. They satisfy the orthogonality relation

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

See also

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References

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  • C. V. L. Charlier (1905–1906) Über die Darstellung willkürlicher Funktionen, Ark. Mat. Astr. och Fysic 2, 20.
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • Szegő, Gabor (1939), Orthogonal Polynomials, Colloquium Publications – American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517