Q梅西納-帕拉澤克多項式

QMEIXNER-POLLACZEK 2D PLOT

Q梅西納-帕拉澤克多項式定義如下：

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

極限關係

連續q哈恩多項式→Q梅西納-帕拉澤克多項式

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

Q梅西納-帕拉澤克多項式→連續q超球面多項式

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

Q梅西納-帕拉澤克多項式→連續q拉蓋爾多項式

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

圖集

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參考文獻

• 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, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248