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Statements
p
n
(
cos
θ
)
=
p
n
(
cos
θ
;
a
,
b
,
c
,
d
|
q
)
=
a
-
n
∑
ℓ
=
0
n
q
ℓ
(
a
b
q
ℓ
,
a
c
q
ℓ
,
a
d
q
ℓ
;
q
)
n
-
ℓ
(
q
-
n
,
a
b
c
d
q
n
-
1
;
q
)
ℓ
(
q
;
q
)
ℓ
∏
j
=
0
ℓ
-
1
(
1
-
2
a
q
j
cos
θ
+
a
2
q
2
j
)
=
a
-
n
(
a
b
,
a
c
,
a
d
;
q
)
n
ϕ
3
4
(
q
-
n
,
a
b
c
d
q
n
-
1
,
a
e
i
θ
,
a
e
-
i
θ
a
b
,
a
c
,
a
d
;
q
,
q
)
.
subscript
𝑝
𝑛
𝜃
Askey-Wilson-polynomial-p
𝑛
𝜃
𝑎
𝑏
𝑐
𝑑
𝑞
superscript
𝑎
𝑛
superscript
subscript
ℓ
0
𝑛
superscript
𝑞
ℓ
q-multiple-Pochhammer
𝑎
𝑏
superscript
𝑞
ℓ
𝑎
𝑐
superscript
𝑞
ℓ
𝑎
𝑑
superscript
𝑞
ℓ
𝑞
𝑛
ℓ
q-multiple-Pochhammer
superscript
𝑞
𝑛
𝑎
𝑏
𝑐
𝑑
superscript
𝑞
𝑛
1
𝑞
ℓ
q-Pochhammer-symbol
𝑞
𝑞
ℓ
superscript
subscript
product
𝑗
0
ℓ
1
1
2
𝑎
superscript
𝑞
𝑗
𝜃
superscript
𝑎
2
superscript
𝑞
2
𝑗
superscript
𝑎
𝑛
q-multiple-Pochhammer
𝑎
𝑏
𝑎
𝑐
𝑎
𝑑
𝑞
𝑛
q-hypergeometric-rphis
4
3
superscript
𝑞
𝑛
𝑎
𝑏
𝑐
𝑑
superscript
𝑞
𝑛
1
𝑎
superscript
𝑒
imaginary-unit
𝜃
𝑎
superscript
𝑒
imaginary-unit
𝜃
𝑎
𝑏
𝑎
𝑐
𝑎
𝑑
𝑞
𝑞
{\displaystyle{\displaystyle p_{n}(\cos\theta)=p_{n}\left(\cos\theta;a,b,c,d\,%
|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{%
\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{\left%
(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2aq^{j}\cos\theta+a^{2}q^{2j})}=a^{%
-n}\left(ab,ac,ad;q\right)_{n}\*{{}_{4}\phi_{3}}\left({q^{-n},abcdq^{n-1},ae^{%
\mathrm{i}\theta},ae^{-\mathrm{i}\theta}\atop ab,ac,ad};q,q\right).}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
(
a
;
q
)
n
q-Pochhammer-symbol
𝑎
𝑞
𝑛
{\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
q-hypergeometric-rphis
𝑟
1
𝑠
subscript
𝑎
0
…
subscript
𝑎
𝑟
subscript
𝑏
1
…
subscript
𝑏
𝑠
𝑞
𝑧
{\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},%
\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
q-multiple-Pochhammer
subscript
𝑎
1
subscript
𝑎
2
…
subscript
𝑎
𝑟
𝑞
𝑛
{\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}%
\right)_{\NVar{n}}}}
q
𝑞
{\displaystyle{\displaystyle q}}
ℓ
ℓ
{\displaystyle{\displaystyle\ell}}
n
𝑛
{\displaystyle{\displaystyle n}}
p
n
(
x
)
subscript
𝑝
𝑛
𝑥
{\displaystyle{\displaystyle p_{n}(x)}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
