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

M n ( x , β , γ ) = ∑ k = 0 n ( − 1 ) k ( n k ) ( x k ) k ! ( x + β ) n − k γ − k {\displaystyle 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

  • Meixner, J. (1934). "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion". Journal of the London Mathematical Society. s1-9: 6–13. doi:.
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  • Andrews, George E.; Askey, Richard (1985). "Classical orthogonal polynomials". Orthogonal polynomials and applications (Bar-le-Duc, 1984). Lecture Notes in Mathematics. Vol.1171. Berlin: Springer. pp.36–62. doi:. ISBN978-3-540-16059-5. MR.
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  • Bavinck, H.; Vanhaeringen, H. (1994). . J. Math. Anal. Appl. 184 (3): 453–463. doi:.
  • Jin, X.-S.; Wong, R. (1998). "Uniform asymptotic expansion for Meixner polynomials". Construct. Approx. 14 (1): 113–150. doi:.
  • Álvarez de Morales, Maria; Pérez, T. E.; Piñar, M. A.; Ronveaux, A. (1999). (PDF). Electron. Trans. Numer. Anal. 9: 1–25. Archived from (PDF) on 2004-09-23.
  • Jin, X.-S.; Wong, R. (1999). . J. Approx. Theory. 96 (2): 281–300. doi:.
  • Borodin, Alexei; Olshanski, Grigori (2006). "Meixner polynomials and random partitions". arXiv:.
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), , in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0-521-19225-5, MR.
  • Boelen, L.; Filipuk, Galina; Van Assche, Walter (2011). "Recurrence coefficients of generalized Meixner polynomials and Peinlevé equations". J. Phys. A: Math. Theor. 44 (3) 035202. Bibcode:. doi:.
  • Wang, Xiang-Sheng; Wong, Roderick (2011). "Global asymptotics of the Meixner polynomials". Asymptot. Anal. 75 (3–4): 211–231. arXiv:. doi:.