连续q哈恩多项式

连续q-哈恩多项式以基本超几何函数定义：^[1]${\displaystyle p_{n}(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}*_{4}\Phi _{3}(q^{-n},abcdq^{n-1},ae^{i{t+2u}},ae^{-it};abe^{2iu},ac,ad;q;q)}$

并且

${\displaystyle x=cos(t+u)}$

极限关系

连续q哈恩多项式→Q梅西纳-帕拉泽克多项式

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

阿斯基-威尔逊多项式→连续q哈恩多项式

在阿斯基-威尔逊多项式中作代换${\displaystyle \theta \to \theta +\phi }$ ,${\displaystyle a\to ae^{i\theta }}$ ,${\displaystyle b\to be^{i\theta }}$ ,${\displaystyle c\to ce^{-i\theta }}$ ,${\displaystyle d\to de^{-i\theta }}$ 即得连续q哈恩多项式： ${\displaystyle p_{n}(cos(\theta +\phi );ae^{i\theta },be^{i\theta },ce^{-i\theta },de^{-i\theta }|q)=p_{n}(cos(\theta +\phi ),a,b,c,d;q)}$ 

图集

CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT

CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT

CONTINUOUS q hahn RE COMPLEX3D Maple PLOT

CONTINUOUS q hahn ABS densitu Maple PLOT

CONTINUOUS q hahn im densitu Maple PLOT

CONTINUOUS q hahn RE densitu 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., Chapter 18: Orthogonal Polynomials, 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, p433