# 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

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

## References

• Meixner, J. (1934). "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion". Journal of the London Mathematical Society s1–9: 6–13. doi:10.1112/jlms/s1-9.1.6.
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• Andrews, G. E.; Askey, Richard (1985). "Classical orthogonal polynomials". Lect. Notes Math. 1171: 36–82. doi:10.1007/BFb0076530.
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• Tratnik, M. V. (1991). "Some multivariable orthogonal polynomials of the Askey tableau-discrete families". J. Math. Phys. 32 (9): 2337. doi:10.1063/1.529158.
• Bavinck, H.; Vanhaeringen, H. (1994). "Difference equations for generalized Meixner Polynomials". J. Math. Anal. Applic. 184 (3): 453–463. doi:10.1006/jmaa.1994.1214.
• Jin, X.-S.; Wong, R. (1998). "Uniform asymptotic expansion for Meixner polynomials". Construct. Approx. 14 (1): 113–150. doi:10.1007/s003659900066.
• Álvarez de Morales, Maria; Pérez, T. E.; Piñar, M. A.; Ronveaux, A. (1999). "Non-standard orthogonality for Meixner Polynomials". El. Trans. Num. Anal. 9: 1–25.
• Jin, X.-S.; Wong, R. (1999). "Asymptotic formulas for the zeros of Meixner Polynomials". J. Approx. Theory 96 (2): 281–300. doi:10.1006/jath.1998.3235.
• Borodin, Alexei; Olshanski, Grigori (2006). "Meixner polynomials and random partitions". arXiv:math/0609806.
• 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., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
• Boelen, L.; Filipuk, Galina; Van Assche, Walter (2011). "Recurrence coefficients of genralized Meixner polynomials and Peinlevé equations". J. Phys. A: Math. Theor. 44 (3): 035202. doi:10.1088/1751-8113/44/3/035202.