梅西纳多项式

梅西纳多项式定义为 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}} 梅西纳多项式的前几项为： M x [ 1 ] := [ − β − ( 1 − γ ) ∗ x / γ ] {\displaystyle Mx[1]:=[-\beta -(1-\gamma )*x/\gamma ]} M x [ 2 ] := [ ( γ 2 ∗ β 2 − β ∗ γ 2 ) / γ 2 + ( γ 2 − 2 ∗ β ∗ γ 2 − 1 + 2 ∗ γ ∗ β ) ∗ x / γ 2 + ( − 2 ∗ γ + 1 + γ 2 ) ∗ x 2 / γ 2 ] {\displaystyle Mx[2]:=[(\gamma ^{2}*\beta ^{2}-\beta *\gamma ^{2})/\gamma ^{2}+(\gamma ^{2}-2*\beta *\gamma ^{2}-1+2*\gamma *\beta )*x/\gamma ^{2}+(-2*\gamma +1+\gamma ^{2})*x^{2}/\gamma ^{2}]} M x [ 3 ] := [ − ( γ 3 ∗ β 3 − 3 ∗ γ 3 ∗ β 2 + 2 ∗ γ 3 ∗ β ) / γ 3 − ( − 3 ∗ γ 3 ∗ β 2 + 3 ∗ γ 2 ∗ β 2 − 3 ∗ γ ∗ β + 6 ∗ γ 3 ∗ β − 2 ∗ γ 3 − 3 ∗ β ∗ γ 2 + 2 ) ∗ x / γ 3 − ( − 3 ∗ γ 3 + 3 ∗ γ 3 ∗ β + 3 ∗ γ 2 − 6 ∗ β ∗ γ 2 + 3 ∗ γ ∗ β − 3 + 3 ∗ γ ) ∗ x 2 / γ 3 − ( 3 ∗ γ 2 − 3 ∗ γ − γ 3 + 1 ) ∗ x 3 / γ 3 ] {\displaystyle Mx[3]:=[-(\gamma ^{3}*\beta ^{3}-3*\gamma ^{3}*\beta ^{2}+2*\gamma ^{3}*\beta )/\gamma ^{3}-(-3*\gamma ^{3}*\beta ^{2}+3*\gamma ^{2}*\beta ^{2}-3*\gamma *\beta +6*\gamma ^{3}*\beta -2*\gamma ^{3}-3*\beta *\gamma ^{2}+2)*x/\gamma ^{3}-(-3*\gamma ^{3}+3*\gamma ^{3}*\beta +3*\gamma ^{2}-6*\beta *\gamma ^{2}+3*\gamma *\beta -3+3*\gamma )*x^{2}/\gamma ^{3}-(3*\gamma ^{2}-3*\gamma -\gamma ^{3}+1)*x^{3}/\gamma ^{3}]} Meixner, J. Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. Journal of the London Mathematical Society. 1934, s1–9: 6–13. doi:10.1112/jlms/s1-9.1.6.  Al-Salam, W. A. On a characterization of Meixner's Polynomials. Quart. J. Math. 1966, 17 (1): 7–10. doi:10.1093/qmath/17.1.7.  Atakishiyev, N. M.; Suslov, S. K. The Hahn and Meixner polynomials of an imaginary argument and some of their applications. J. Phys. A: Math. Gen. 1985, 18 (10): 1583. doi:10.1088/0305-4470/18/10/014.  Andrews, G. E.; Askey, Richard. Classical orthogonal polynomials. Lect. Notes Math. 1985, 1171: 36–82. doi:10.1007/BFb0076530.  Tratnik, M. V. Multivariable Meixer, Krawtchouk, and Meixner-Pollaczek polynomials. J. Math. Phys. 1989, 30 (12): 2740. doi:10.1063/1.528507.  Tratnik, M. V. Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 1991, 32 (9): 2337. doi:10.1063/1.529158.  Bavinck, H.; Vanhaeringen, H. Difference equations for generalized Meixner Polynomials. J. Math. Anal. Applic. 1994, 184 (3): 453–463. doi:10.1006/jmaa.1994.1214.  Jin, X.-S.; Wong, R. Uniform asymptotic expansion for Meixner polynomials. Construct. Approx. 1998, 14 (1): 113–150. doi:10.1007/s003659900066.  Álvarez de Morales, Maria; Pérez, T. E.; Piñar, M. A.; Ronveaux, A. Non-standard orthogonality for Meixner Polynomials (PDF). El. Trans. Num. Anal. 1999, 9: 1–25 [2015-01-27]. （原始内容 (PDF)存档于2008-11-22）.  Jin, X.-S.; Wong, R. Asymptotic formulas for the zeros of Meixner Polynomials. J. Approx. Theory. 1999, 96 (2): 281–300. doi:10.1006/jath.1998.3235.  Borodin, Alexei; Olshanski, Grigori. Meixner polynomials and random partitions. 2006. arXiv:math/0609806 .  Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Hahn Class: Definitions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248  Boelen, L.; Filipuk, Galina; Van Assche, Walter. Recurrence coefficients of genralized Meixner polynomials and Peinlevé equations. J. Phys. A: Math. Theor. 2011, 44 (3): 035202. doi:10.1088/1751-8113/44/3/035202.