查看“︁阿斯基-威尔逊多项式”︁的源代码

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[[File:Askey-Wilson polynomials 1.gif|thumb|2nd  order  Askey-Wilson polynomials]]
[[File:Askey-Wilson polynomials 2.gif|thumb|2nd  order  Askey-Wilson polynomials]]

'''阿斯基-威尔逊多项式'''（Askey–Wilson polynomials）是一个以[[基本超几何函数]]表示的[[正交多项式]]:

<math>\frac{a^{n}p_{n}(x;a,b,c,d,|q)}{(ab,ac,ad;q)_{n}}=_4\phi_{3}(q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta};ab,ac,ad;q,q)</math>

其中<math>x=cos(\theta)</math>
阿斯基-威尔逊多项式是[[威尔逊多项式]]的[[q模拟]].

==极限关系==
;[[阿斯基-威尔逊多项式]]→[[连续双q哈恩多项式]]
在阿斯基-威尔逊多项式中，令<math>d=0</math>即得连续双哈恩多项式<ref name=K>Roelof,p420</ref>
<math>p_{n}(x;a,b,c,0|q)=p_{n}(x;a,b,c|q)</math>

;[[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<math>\theta  \to  \theta+\phi</math>,<math>a \to ae^{i\theta}</math>,<math>b \to be^{i\theta}</math>,<math>c \to ce^{-i\theta}</math>,<math>d \to de^{-i\theta}</math>即得连续q哈恩多项式：
<math>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)</math>

;[[阿斯基-威尔逊多项式]]→[[大q雅可比多项式]]

==参考文献==
<references/>
Roelof KoekoeK,Peter Lesky  Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer 2010*{{Citation | authorlink=Richard Askey | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials | isbn=978-0-8218-2321-7 | mr=783216 | year=1985 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=54 | issue=319 | pages=iv+55|url=http://books.google.com/books?id=9q9o03nD_xsC | doi=10.1090/memo/0319}}
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
*{{dlmf|id=18.28|title=Askey-Wilson class|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | first=Tom H. | last=Koornwinder | title=[http://www.scholarpedia.org/article/Askey-Wilson_polynomial Askey-Wilson polynomial] | journal=Scholarpedia | volume=7 | year=2012 | issue=7 | pages=7761 | doi=10.4249/scholarpedia.7761}}

[[Category:正交多项式]]
[[Category:基本超几何函数]]