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Statements
C
n
(
λ
)
(
cos
θ
1
)
C
n
(
λ
)
(
1
)
C
n
(
λ
)
(
cos
θ
2
)
C
n
(
λ
)
(
1
)
=
Γ
(
λ
+
1
2
)
π
1
2
Γ
(
λ
)
∫
0
π
C
n
(
λ
)
(
cos
θ
1
cos
θ
2
+
sin
θ
1
sin
θ
2
cos
ϕ
)
C
n
(
λ
)
(
1
)
(
sin
ϕ
)
2
λ
-
1
d
ϕ
,
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
subscript
𝜃
1
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
1
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
subscript
𝜃
2
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
1
Euler-Gamma
𝜆
1
2
superscript
𝜋
1
2
Euler-Gamma
𝜆
superscript
subscript
0
𝜋
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
subscript
𝜃
1
subscript
𝜃
2
subscript
𝜃
1
subscript
𝜃
2
italic-ϕ
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
1
superscript
italic-ϕ
2
𝜆
1
italic-ϕ
{\displaystyle{\displaystyle\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)%
}{C^{(\lambda)}_{n}\left(1\right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}%
\right)}{C^{(\lambda)}_{n}\left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2%
}\right)}{\pi^{\frac{1}{2}}\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^%
{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}%
\cos\phi\right)}{C^{(\lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}%
\mathrm{d}\phi,}}
λ
>
0
𝜆
0
{\displaystyle{\displaystyle\lambda>0}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\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
