Meixner polynomials

Not to be confused with Meixner–Pollaczek polynomials.

In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (1934). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by

$M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x-\beta)_{n-k}\gamma^{-k}$

See also

• Kravchuk polynomials

References

• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/18.19 
• Meixner, J. (1934), "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion", J. London Math. Soc. s1-9: 6–13, doi:10.1112/jlms/s1-9.1.6