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Statements
𝖯𝗌
n
m
(
x
,
γ
2
)
=
∑
k
=
-
R
∞
(
-
1
)
k
a
n
,
k
m
(
γ
2
)
𝖯
n
+
2
k
m
(
x
)
,
spheroidal-wave-on-cut-Ps
𝑚
𝑛
𝑥
superscript
𝛾
2
superscript
subscript
𝑘
𝑅
superscript
1
𝑘
subscript
superscript
𝑎
𝑚
𝑛
𝑘
superscript
𝛾
2
Ferrers-Legendre-P-first-kind
𝑚
𝑛
2
𝑘
𝑥
{\displaystyle{\displaystyle\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_%
{k=-R}^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x%
\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)}}
𝖯𝗌
n
m
(
x
,
γ
2
)
spheroidal-wave-on-cut-Ps
𝑚
𝑛
𝑥
superscript
𝛾
2
{\displaystyle{\displaystyle\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},%
\NVar{\gamma^{2}}\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
≥
m
𝑛
𝑚
{\displaystyle{\displaystyle n\geq m}}
γ
2
superscript
𝛾
2
{\displaystyle{\displaystyle\gamma^{2}}}
R
=
⌊
(
n
-
m
)
/
2
⌋
𝑅
𝑛
𝑚
2
{\displaystyle{\displaystyle R=\left\lfloor(n-m)/2\right\rfloor}}
a
n
,
k
m
(
γ
2
)
subscript
superscript
𝑎
𝑚
𝑛
𝑘
superscript
𝛾
2
{\displaystyle{\displaystyle a^{m}_{n,k}(\gamma^{2})}}
