Meixner 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

[math]\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} }[/math]

See also

• Kravchuk polynomials

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. 
• 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. Bibcode: 1966QJMat..17....7A. 
• 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. Bibcode: 1985JPhA...18.1583A. 
• Andrews, George E.; Askey, Richard (1985). "Orthogonal polynomials and applications (Bar-le-Duc, 1984)". 1171. Berlin: Springer. pp. 36–62. 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. Bibcode: 1989JMP....30.2740T. https://zenodo.org/record/1232297. 
• Tratnik, M. V. (1991). "Some multivariable orthogonal polynomials of the Askey tableau-discrete families". J. Math. Phys. 32 (9): 2337–2342. doi:10.1063/1.529158. Bibcode: 1991JMP....32.2337T. https://zenodo.org/record/1232107. 
• Bavinck, H.; Vanhaeringen, H. (1994). "Difference equations for generalized Meixner Polynomials". J. Math. Anal. Appl. 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". Electron. Trans. Numer. Anal. 9: 1–25. http://www.emis.de/journals/ETNA/vol.9.1999/pp1-25.dir/pp1-25.pdf. Retrieved 2013-03-10. 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18.19 
• 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. doi:10.1088/1751-8113/44/3/035202. Bibcode: 2011JPhA...44c5202B. 
• Wang, Xiang-Sheng; Wong, Roderick (2011). "Global asymptotics of the Meixner polynomials". Asymptot. Anal. 75 (3–4): 211–231. doi:10.3233/ASY-2011-1060.