Meixner polynomials

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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[edit]


  • 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. 
  • Al-Salam, W. A. (1966). "On a characterization of Meixner's Polynomials". Quart. J. Math. 17 (1): 7–10. doi:10.1093/qmath/17.1.7. 
  • Atakishiyev, N. M.; Suslov, S. K. (1985). "The Hahn and Meixner polynomials of an imaginary argument and some of their applications". J. Phys. A: Math. Gen. 18 (10): 1583. doi:10.1088/0305-4470/18/10/014. 
  • Andrews, G. E.; Askey, Richard (1985). "Classical orthogonal polynomials". Lect. Notes Math. 1171: 36–82. doi:10.1007/BFb0076530. 
  • Tratnik, M. V. (1989). "Multivariable Meixer, Krawtchouk, and Meixner-Pollaczek polynomials". J. Math. Phys. 30 (12): 2740. doi:10.1063/1.528507. 
  • 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. 
  • Olshanski, Alexei (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.