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Statements
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𝖰
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𝖯
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𝖰
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𝖯
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ν
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,
𝑥
superscript
1
superscript
𝑥
2
3
2
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
𝑥
1
1
4
superscript
𝜇
2
superscript
1
superscript
𝑥
2
1
2
1
2
superscript
𝜇
2
2
𝜈
𝜈
1
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
2
𝜈
1
𝜇
𝜈
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
2
superscript
𝜇
𝜈
1
2
Ferrers-Legendre-P-first-kind
𝜇
𝜈
1
𝑥
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
1
𝑥
{\displaystyle{\displaystyle\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\mathsf{P}%
^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=%
\frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2%
\nu(\nu+1))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x%
\right)+(2\nu+1)(\mu-\nu-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))+2(\mu-\nu-1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x%
\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right),}}
μ
≠
±
1
2
𝜇
plus-or-minus
1
2
{\displaystyle{\displaystyle\mu\neq\pm\frac{1}{2}}}
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}}
Identifiers
