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
M
n
(
x
,
β β -->
,
γ γ -->
)
=
∑ ∑ -->
k
=
0
n
(
− − -->
1
)
k
(
n
k
)
(
x
k
)
k
!
(
x
+
β β -->
)
n
− − -->
k
γ 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
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, 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 :10.1007/BFb0076530 . MR 0838970 . 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) . Electron. Trans. Numer. Anal . 9 : 1– 25. Archived from the original (PDF) on 2004-09-23. 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 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 .
