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. They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by

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

Meixner polynomials belong to the Hahn class of polynomials, along with Hahn polynomials, Charlier polynomials, and Kravchuk polynomials.

See also

• Kravchuk polynomials

References

• Al-Salam, W. A. (1966). "On a characterization of Meixner's Polynomials". Quarterly Journal of Mathematics 17 (1): 7–10. doi:10.1093/qmath/17.1.7. 

• Álvarez de Morales, Maria; Pérez, Teresa E.; Piñar, Miguel A.; Ronveaux, André (1999). "Non-standard orthogonality for Meixner Polynomials". Electronic Transactions on Numerical Analysis 9: 1–25. https://web.archive.org/web/20040923194437/http://www.math.ethz.ch/EMIS/journals/ETNA/vol.9.1999/pp1-25.dir/pp1-25.pdf. 

• Andrews, George E.; Askey, Richard (1985). "Polynômes Orthogonaux et Applications". in Brezinski, Claude; Draux, André; Magnus, Alphonse P. et al.. Bar-le-Duc, 1984. 1171. Berlin, Heidelberg: Springer. pp. 36–62. doi:10.1007/BFb0076530. ISBN 978-3-540-16059-5. 

• Atakishiyev, N. M.; Suslov, S. K. (1985). "The Hahn and Meixner polynomials of an imaginary argument and some of their applications". Journal of Physics A: Mathematical and General 18 (10): 1583–1596. doi:10.1088/0305-4470/18/10/014. 

• Bavinck, Herman; van Haeringen, Henk (1994). "Difference equations for generalized Meixner polynomials". Journal of Mathematical Analysis and Applications 184 (3): 453–463. doi:10.1006/jmaa.1994.1214. 

• Boelen, Lies; Filipuk, Galina; Van Assche, Walter (2011). "Recurrence coefficients of generalized Meixner polynomials and Painlevé equations". Journal of Physics A: Mathematical and Theoretical 44 (3). doi:10.1088/1751-8113/44/3/035202. 

• Borodin, Alexei; Olshanski, Grigori (2006). "Meixner polynomials and random partitions". Moscow Mathematical Journal 6 (4): 629–655. doi:10.17323/1609-4514-2006-6-4-629-655. 

• Jin, X.-S.; Wong, R. (1998). "Uniform asymptotic expansion for Meixner polynomials". Constructive Approximation 14 (1): 113–150. doi:10.1007/s003659900066. 

• Jin, X.-S.; Wong, R. (1999). "Asymptotic formulas for the zeros of Meixner polynomials". Journal of Approximation Theory 96 (2): 281–300. doi:10.1006/jath.1998.3235. 

• Koornwinder, T. H.; Wong, R.; Koekoek, R.; Swarttouw, R. F. (2010). "Orthogonal polynomials". in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al.. NIST Handbook of Mathematical Functions. Cambridge University Press. pp. 435–484. ISBN 978-0-521-19225-5. https://dlmf.nist.gov/18.19. 

• Meixner, J. (1934). "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion" (in de). Journal of the London Mathematical Society. 1 9 (1): 6–13. doi:10.1112/jlms/s1-9.1.6. 

• Tratnik, M. V. (1989). "Multivariable Meixer, Krawtchouk, and Meixner-Pollaczek polynomials". Journal of Mathematical Physics 30 (12): 2740–2749. doi:10.1063/1.528507. https://zenodo.org/record/1232297. 

• Tratnik, M. V. (1991). "Some multivariable orthogonal polynomials of the Askey tableau-discrete families". Journal of Mathematical Physics 32 (9): 2337–2342. doi:10.1063/1.529158. https://zenodo.org/record/1232107. 

• Wang, X.-S.; Wong, R. (2011). "Global asymptotics of the Meixner polynomials". Asymptotic Analysis 75 (3–4): 211–231. doi:10.3233/ASY-2011-1060.