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