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Statements
∫
x
𝖯
ν
μ
(
x
)
𝖰
ν
μ
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x
)
d
x
=
1
2
ν
(
ν
+
1
)
(
(
μ
2
-
(
ν
+
1
)
(
ν
+
x
2
)
)
𝖯
ν
μ
(
x
)
𝖰
ν
μ
(
x
)
+
(
ν
+
1
)
(
ν
-
μ
+
1
)
x
(
𝖯
ν
μ
(
x
)
𝖰
ν
+
1
μ
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+
𝖯
ν
+
1
μ
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x
)
𝖰
ν
μ
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x
)
)
-
(
ν
-
μ
+
1
)
2
𝖯
ν
+
1
μ
(
x
)
𝖰
ν
+
1
μ
(
x
)
)
,
𝑥
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
𝑥
1
2
𝜈
𝜈
1
superscript
𝜇
2
𝜈
1
𝜈
superscript
𝑥
2
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
𝜈
1
𝜈
𝜇
1
𝑥
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
1
𝑥
Ferrers-Legendre-P-first-kind
𝜇
𝜈
1
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
superscript
𝜈
𝜇
1
2
Ferrers-Legendre-P-first-kind
𝜇
𝜈
1
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
1
𝑥
{\displaystyle{\displaystyle\int x\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{%
Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(%
\nu+1)(\nu+x^{2}))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}%
\left(x\right)+(\nu+1)(\nu-\mu+1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf%
{Q}^{\mu}_{\nu+1}\left(x\right)+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{%
Q}^{\mu}_{\nu}\left(x\right))-(\nu-\mu+1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x%
\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right),}}
ν
≠
0
,
-
1
𝜈
0
1
{\displaystyle{\displaystyle\nu\neq 0,-1}}
𝖯
ν
μ
(
x
)
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
𝖰
ν
μ
(
x
)
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
