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

$M_n(x,\beta ,\gamma )=\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})(\genfrac{}{}{0ex}{}{x}{k})k!(x+\beta {)}_{n-k}{\gamma }^{-k}$

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. Bibcode:1966QJMat..17....7A. 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. Bibcode:1985JPhA...18.1583A. doi:10.1088/0305-4470/18/10/014.
• Andrews, G. E.; Askey, Richard (1985). "Classical orthogonal polynomials". Lect. Notes Math. Lecture Notes in Mathematics. 1171: 36–82. doi:10.1007/BFb0076530. ISBN 978-3-540-16059-5.
• Tratnik, M. V. (1989). "Multivariable Meixer, Krawtchouk, and Meixner-Pollaczek polynomials". J. Math. Phys. 30 (12): 2740. Bibcode:1989JMP....30.2740T. doi:10.1063/1.528507.
• Tratnik, M. V. (1991). "Some multivariable orthogonal polynomials of the Askey tableau-discrete families". J. Math. Phys. 32 (9): 2337–2342. Bibcode:1991JMP....32.2337T. doi:10.1063/1.529158.
• 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" (PDF). El. Trans. Num. Anal. 9: 1–25. Archived from the original (PDF) on 2008-11-22. 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.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• 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:2011JPhA...44c5202B. doi:10.1088/1751-8113/44/3/035202.
• Wang, Xiang-Sheng; Wong, Roderick (2011). "Global asymptotics of the Meixner polynomials". Asymptot. Anal. 75 (3–4): 211–231. arXiv:1101.4370. doi:10.3233/ASY-2011-1060.