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Statements
P
n
-
1
2
m
(
cosh
ξ
)
=
Γ
(
n
+
m
+
1
2
)
Γ
(
n
-
m
+
1
2
)
(
2
π
sinh
ξ
)
1
/
2
𝑸
m
-
1
2
n
(
coth
ξ
)
,
Legendre-P-first-kind
𝑚
𝑛
1
2
𝜉
Euler-Gamma
𝑛
𝑚
1
2
Euler-Gamma
𝑛
𝑚
1
2
superscript
2
𝜋
𝜉
1
2
associated-Legendre-black-Q
𝑛
𝑚
1
2
hyperbolic-cotangent
𝜉
{\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{%
\Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*%
\left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}%
\left(\coth\xi\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)}}
𝑸
ν
μ
(
z
)
associated-Legendre-black-Q
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(%
\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
coth
z
hyperbolic-cotangent
𝑧
{\displaystyle{\displaystyle\coth\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
ξ
>
0
𝜉
0
{\displaystyle{\displaystyle\xi>0}}
