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Statements
P
ν
-
1
2
μ
(
cosh
ξ
)
=
Γ
(
1
2
-
μ
)
π
1
/
2
(
1
-
e
-
2
ξ
)
μ
e
(
ν
+
(
1
/
2
)
)
ξ
𝐅
(
1
2
-
μ
,
1
2
+
ν
-
μ
;
1
-
2
μ
;
1
-
e
-
2
ξ
)
,
Legendre-P-first-kind
𝜇
𝜈
1
2
𝜉
Euler-Gamma
1
2
𝜇
superscript
𝜋
1
2
superscript
1
superscript
𝑒
2
𝜉
𝜇
superscript
𝑒
𝜈
1
2
𝜉
scaled-hypergeometric-bold-F
1
2
𝜇
1
2
𝜈
𝜇
1
2
𝜇
1
superscript
𝑒
2
𝜉
{\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=%
\frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{%
\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu%
;1-2\mu;1-e^{-2\xi}\right),}}
μ
≠
1
2
,
3
2
,
5
2
,
…
𝜇
1
2
3
2
5
2
…
{\displaystyle{\displaystyle\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots}}
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}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\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)}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
ξ
>
0
𝜉
0
{\displaystyle{\displaystyle\xi>0}}
Identifiers
