离散q-埃尔米特I多项式

离散q-埃尔米特多项式是以超几何函数定义的正交多项式[1]

h n ( x ; q ) = q ( n 2 ) 2 ϕ 1 ( q − n , x − 1 ; 0 ; q , − q x ) = x n 2 ϕ 0 ( q − n , q − n + 1 ; ; q 2 , q 2 n − 1 / x 2 ) = U n ( − 1 ) ( x ; q ) {\displaystyle \displaystyle h_{n}(x;q)=q^{\binom {n}{2}}{}_{2}\phi _{1}(q^{-n},x^{-1};0;q,-qx)=x^{n}{}_{2}\phi _{0}(q^{-n},q^{-n+1};;q^{2},q^{2n-1}/x^{2})=U_{n}^{(-1)}(x;q)}

图集

DISCRETE Q-HERMITE I ABS COMPLEX 3D MAPLE PLOT
DISCRETE Q-HERMITE I IM COMPLEX 3D MAPLE PLOT
DISCRETE Q-HERMITE I RE COMPLEX 3D MAPLE PLOT
DISCRETE Q-HERMITE I ABS DENSITY MAPLE PLOT
DISCRETE Q-HERMITE I IM DENSITY MAPLE PLOT
DISCRETE Q-HERMITE I RE DENSITY MAPLE PLOT

参考文献

  1. ^ Roelof Koekoek, p547-549,Springer 2010
  • Berg, Christian; Ismael, Mourad, Q-Hermite Polynomials and Classical Orthogonal Polynomials [1], 1994  外部链接存在于|title= (帮助)
  • Al-Salam, W. A.; Carlitz, L., Some orthogonal q-polynomials, Mathematische Nachrichten, 1965, 30: 47–61, ISSN 0025-584X, MR 0197804, doi:10.1002/mana.19650300105 
  • Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5 
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18 |contribution-url=缺少标题 (帮助), 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 
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q超几何函数与q超几何正交多项式
q超几何函数q超几何正交多项式