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Statements
P
n
(
α
,
β
)
(
cos
θ
)
P
n
(
α
,
β
)
(
1
)
=
2
Γ
(
α
+
1
)
π
1
2
Γ
(
α
-
β
)
Γ
(
β
+
1
2
)
∫
0
1
∫
0
π
(
(
cos
1
2
θ
)
2
-
r
2
(
sin
1
2
θ
)
2
+
i
r
sin
θ
cos
ϕ
)
n
(
1
-
r
2
)
α
-
β
-
1
r
2
β
+
1
(
sin
ϕ
)
2
β
d
ϕ
d
r
,
Jacobi-polynomial-P
𝛼
𝛽
𝑛
𝜃
Jacobi-polynomial-P
𝛼
𝛽
𝑛
1
2
Euler-Gamma
𝛼
1
superscript
𝜋
1
2
Euler-Gamma
𝛼
𝛽
Euler-Gamma
𝛽
1
2
superscript
subscript
0
1
superscript
subscript
0
𝜋
superscript
superscript
1
2
𝜃
2
superscript
𝑟
2
superscript
1
2
𝜃
2
𝑖
𝑟
𝜃
italic-ϕ
𝑛
superscript
1
superscript
𝑟
2
𝛼
𝛽
1
superscript
𝑟
2
𝛽
1
superscript
italic-ϕ
2
𝛽
italic-ϕ
𝑟
{\displaystyle{\displaystyle\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right%
)}{P^{(\alpha,\beta)}_{n}\left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{%
\pi^{\frac{1}{2}}\Gamma\left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}%
\right)}\*\int_{0}^{1}\int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(%
\sin\tfrac{1}{2}\theta)^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-%
\beta-1}r^{2\beta+1}(\sin\phi)^{2\beta}\mathrm{d}\phi\mathrm{d}r,}}
α
>
β
>
-
1
2
𝛼
𝛽
1
2
{\displaystyle{\displaystyle\alpha>\beta>-\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}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
n
𝑛
{\displaystyle{\displaystyle n}}
