DLMF:14.17.E5 (Q4886)

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DLMF:14.17.E5
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Statements

${\displaystyle{\displaystyle\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}% \mathsf{P}^{-\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{\Gamma\left(\frac{1}{2}% \sigma+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\sigma+1\right)}{2^{\mu+1}% \Gamma\left(\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)\Gamma% \left(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}\right)},}}$
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DLMF:14.17.E5
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${\displaystyle{\displaystyle\Re\sigma>-1}}$
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${\displaystyle{\displaystyle\Re\mu>-1}}$
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${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}$
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${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$
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${\displaystyle{\displaystyle\int}}$
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${\displaystyle{\displaystyle\Re}}$
${\displaystyle{\displaystyle x}}$
${\displaystyle{\displaystyle\mu}}$
${\displaystyle{\displaystyle\nu}}$