阿斯基-威爾遜多項式 （Askey–Wilson polynomials）是一個以基本超幾何函數 表示的正交多項式 :
2nd order Askey-Wilson polynomials 2nd order Askey-Wilson polynomials
a
n
p
n
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;
a
,
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,
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,
d
,
|
q
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(
a
b
,
a
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,
a
d
;
q
)
n
=
4
ϕ
3
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q
−
n
,
a
b
c
d
q
n
−
1
,
a
e
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θ
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a
e
−
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)
{\displaystyle {\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)}
其中
x
=
c
o
s
(
θ
)
{\displaystyle x=cos(\theta )}
威爾遜多項式 的q模擬 .
阿斯基-威爾遜多項式 →連續雙q哈恩多項式 在阿斯基-威爾遜多項式中，令
d
=
0
{\displaystyle d=0}
[ 1]
p
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0
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=
p
n
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{\displaystyle p_{n}(x;a,b,c,0|q)=p_{n}(x;a,b,c|q)}
阿斯基-威爾遜多項式 →連續q哈恩多項式 在阿斯基-威爾遜多項式中作代換
θ
→
θ
+
ϕ
{\displaystyle \theta \to \theta +\phi }
a
→
a
e
i
θ
{\displaystyle a\to ae^{i\theta }}
b
→
b
e
i
θ
{\displaystyle b\to be^{i\theta }}
c
→
c
e
−
i
θ
{\displaystyle c\to ce^{-i\theta }}
d
→
d
e
−
i
θ
{\displaystyle d\to de^{-i\theta }}
p
n
(
c
o
s
(
θ
+
ϕ
)
;
a
e
i
θ
,
b
e
i
θ
,
c
e
−
i
θ
,
d
e
−
i
θ
|
q
)
=
p
n
(
c
o
s
(
θ
+
ϕ
)
,
a
,
b
,
c
,
d
;
q
)
{\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)}
阿斯基-威爾遜多項式 →大q雅可比多項式
Roelof KoekoeK,Peter Lesky Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer 2010*Askey, Richard ; Wilson, James, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , Memoirs of the American Mathematical Society, 1985, 54 (319): iv+55, ISBN 978-0-8218-2321-7ISSN 0065-9266 MR 0783216 doi:10.1090/memo/0319
Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press , 2004, ISBN 978-0-521-83357-8MR 2128719 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Askey-Wilson class , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Koornwinder, Tom H., Askey-Wilson polynomial , Scholarpedia, 2012, 7 (7): 7761, doi:10.4249/scholarpedia.7761 
