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Statements
𝖯
ν
(
cos
θ
1
cos
θ
2
+
sin
θ
1
sin
θ
2
cos
ϕ
)
=
𝖯
ν
(
cos
θ
1
)
𝖯
ν
(
cos
θ
2
)
+
2
∑
m
=
1
∞
(
-
1
)
m
𝖯
ν
-
m
(
cos
θ
1
)
𝖯
ν
m
(
cos
θ
2
)
cos
(
m
ϕ
)
,
shorthand-Ferrers-Legendre-P-first-kind
𝜈
subscript
𝜃
1
subscript
𝜃
2
subscript
𝜃
1
subscript
𝜃
2
italic-ϕ
shorthand-Ferrers-Legendre-P-first-kind
𝜈
subscript
𝜃
1
shorthand-Ferrers-Legendre-P-first-kind
𝜈
subscript
𝜃
2
2
superscript
subscript
𝑚
1
superscript
1
𝑚
Ferrers-Legendre-P-first-kind
𝑚
𝜈
subscript
𝜃
1
Ferrers-Legendre-P-first-kind
𝑚
𝜈
subscript
𝜃
2
𝑚
italic-ϕ
{\displaystyle{\displaystyle\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}%
+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_%
{1}\right)\mathsf{P}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)%
^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(%
\cos\theta_{2}\right)\cos\left(m\phi\right),}}
𝖯
ν
μ
(
x
)
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
𝖯
ν
(
x
)
=
𝖯
ν
0
(
x
)
shorthand-Ferrers-Legendre-P-first-kind
𝜈
𝑥
Ferrers-Legendre-P-first-kind
0
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{P}^{0}_{\nu}\left(x\right)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
m
𝑚
{\displaystyle{\displaystyle m}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
θ
1
subscript
𝜃
1
{\displaystyle{\displaystyle\theta_{1}}}
ϕ
italic-ϕ
{\displaystyle{\displaystyle\phi}}
