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Statements
C
n
(
cos
θ
;
β
|
q
)
=
∑
ℓ
=
0
n
(
β
;
q
)
ℓ
(
β
;
q
)
n
-
ℓ
(
q
;
q
)
ℓ
(
q
;
q
)
n
-
ℓ
e
i
(
n
-
2
ℓ
)
θ
=
(
β
;
q
)
n
(
q
;
q
)
n
e
i
n
θ
ϕ
1
2
(
q
-
n
,
β
β
-
1
q
1
-
n
;
q
,
β
-
1
q
e
-
2
i
θ
)
.
q-ultraspherical-polynomial
𝑛
𝜃
𝛽
𝑞
superscript
subscript
ℓ
0
𝑛
q-Pochhammer-symbol
𝛽
𝑞
ℓ
q-Pochhammer-symbol
𝛽
𝑞
𝑛
ℓ
q-Pochhammer-symbol
𝑞
𝑞
ℓ
q-Pochhammer-symbol
𝑞
𝑞
𝑛
ℓ
superscript
𝑒
imaginary-unit
𝑛
2
ℓ
𝜃
q-Pochhammer-symbol
𝛽
𝑞
𝑛
q-Pochhammer-symbol
𝑞
𝑞
𝑛
superscript
𝑒
imaginary-unit
𝑛
𝜃
q-hypergeometric-rphis
2
1
superscript
𝑞
𝑛
𝛽
superscript
𝛽
1
superscript
𝑞
1
𝑛
𝑞
superscript
𝛽
1
𝑞
superscript
𝑒
2
imaginary-unit
𝜃
{\displaystyle{\displaystyle C_{n}\left(\cos\theta;\beta\,|\,q\right)=\sum_{%
\ell=0}^{n}\frac{\left(\beta;q\right)_{\ell}\left(\beta;q\right)_{n-\ell}}{%
\left(q;q\right)_{\ell}\left(q;q\right)_{n-\ell}}e^{\mathrm{i}(n-2\ell)\theta}%
=\frac{\left(\beta;q\right)_{n}}{\left(q;q\right)_{n}}e^{\mathrm{i}n\theta}{{}%
_{2}\phi_{1}}\left({q^{-n},\beta\atop\beta^{-1}q^{1-n}};q,\beta^{-1}qe^{-2%
\mathrm{i}\theta}\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)}}
q
𝑞
{\displaystyle{\displaystyle q}}
ℓ
ℓ
{\displaystyle{\displaystyle\ell}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
