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Statements
P
ν
-
μ
(
x
)
=
π
(
x
2
-
1
)
μ
/
2
2
μ
(
𝐅
(
1
2
μ
-
1
2
ν
,
1
2
ν
+
1
2
μ
+
1
2
;
1
2
;
x
2
)
Γ
(
1
2
μ
-
1
2
ν
+
1
2
)
Γ
(
1
2
ν
+
1
2
μ
+
1
)
-
x
𝐅
(
1
2
μ
-
1
2
ν
+
1
2
,
1
2
ν
+
1
2
μ
+
1
;
3
2
;
x
2
)
Γ
(
1
2
μ
-
1
2
ν
)
Γ
(
1
2
ν
+
1
2
μ
+
1
2
)
)
,
Legendre-P-first-kind
𝜇
𝜈
𝑥
𝜋
superscript
superscript
𝑥
2
1
𝜇
2
superscript
2
𝜇
scaled-hypergeometric-bold-F
1
2
𝜇
1
2
𝜈
1
2
𝜈
1
2
𝜇
1
2
1
2
superscript
𝑥
2
Euler-Gamma
1
2
𝜇
1
2
𝜈
1
2
Euler-Gamma
1
2
𝜈
1
2
𝜇
1
𝑥
scaled-hypergeometric-bold-F
1
2
𝜇
1
2
𝜈
1
2
1
2
𝜈
1
2
𝜇
1
3
2
superscript
𝑥
2
Euler-Gamma
1
2
𝜇
1
2
𝜈
Euler-Gamma
1
2
𝜈
1
2
𝜇
1
2
{\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}%
-1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}%
{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{%
\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{%
1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{%
1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}%
{\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+%
\frac{1}{2}\mu+\frac{1}{2}\right)}\right),}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
𝐅
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
