Q梅西纳-帕拉泽克多项式

Q梅西纳-帕拉泽克多项式定义如下：^[1]

QMEIXNER-POLLACZEK 2D PLOT

${\displaystyle P_{n}(x;a|q)=a^{-n}e^{in\phi }}$${\displaystyle {\frac {a^{2};q_{n}}{(q;q)_{n}}}}$${\displaystyle _{3}\Phi _{2}(q^{-}n,ae^{i(\theta +2\phi )},ae^{-i\theta };a^{2},0|q;q)}$

极限关系

连续q哈恩多项式→Q梅西纳-帕拉泽克多项式

${\displaystyle {\frac {p_{n}(cos(\theta +\phi );a,0,0,a;q)}{(q;q)_{n}}}=P_{n}(cos(\theta +\phi );a|q)}$ 

Q梅西纳-帕拉泽克多项式→连续q超球面多项式

${\displaystyle P_{n}(cos\phi ;\beta |q)=C_{n}(cos\phi );\beta |q)}$ 

Q梅西纳-帕拉泽克多项式→连续q拉盖尔多项式

${\displaystyle P_{n}(cos(\theta +\phi );q^{\alpha /2+1/2}|q)=}$ ${\displaystyle q^{(-\alpha /2-1/4)*n}*P_{n}^{(}\alpha )(cos\theta |q)}$ 

图集

QMEIXNER-POLLACZEK ABS COMPLEX 3D MAPLE PLOT

QMEIXNER-POLLACZEK IM COMPLEX 3D MAPLE PLOT

QMEIXNER-POLLACZEK RE COMPLEX 3D MAPLE PLOT

QMEIXNER-POLLACZEK ABS DENSITY MAPLE PLOT

QMEIXNER-POLLACZEK IM DENSITY MAPLE PLOT

QMEIXNER-POLLACZEK RE DENSITY MAPLE PLOT

参考文献

• 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 

1. Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p460,Springer