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Statements
P
n
(
α
,
α
)
(
cos
θ
)
P
n
(
α
,
α
)
(
1
)
=
C
n
(
α
+
1
2
)
(
cos
θ
)
C
n
(
α
+
1
2
)
(
1
)
=
2
α
+
1
2
Γ
(
α
+
1
)
π
1
2
Γ
(
α
+
1
2
)
(
sin
θ
)
-
2
α
∫
0
θ
cos
(
(
n
+
α
+
1
2
)
ϕ
)
(
cos
ϕ
-
cos
θ
)
-
α
+
1
2
d
ϕ
,
Jacobi-polynomial-P
𝛼
𝛼
𝑛
𝜃
Jacobi-polynomial-P
𝛼
𝛼
𝑛
1
ultraspherical-Gegenbauer-polynomial
𝛼
1
2
𝑛
𝜃
ultraspherical-Gegenbauer-polynomial
𝛼
1
2
𝑛
1
superscript
2
𝛼
1
2
Euler-Gamma
𝛼
1
superscript
𝜋
1
2
Euler-Gamma
𝛼
1
2
superscript
𝜃
2
𝛼
superscript
subscript
0
𝜃
𝑛
𝛼
1
2
italic-ϕ
superscript
italic-ϕ
𝜃
𝛼
1
2
italic-ϕ
{\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta%
\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}%
_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac%
{2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma%
\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{%
\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+%
\frac{1}{2}}}\mathrm{d}\phi,}}
α
>
-
1
2
𝛼
1
2
{\displaystyle{\displaystyle\alpha>-\tfrac{1}{2}}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
n
(
α
,
β
)
(
x
)
Jacobi-polynomial-P
𝛼
𝛽
𝑛
𝑥
{\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(%
\NVar{x}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
C
n
(
λ
)
(
x
)
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝑥
{\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
n
𝑛
{\displaystyle{\displaystyle n}}
Identifiers
