# DLMF:14.3.E16 (Q4705)

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DLMF:14.3.E16
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## Statements

${\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right% )=\frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(% \nu+\mu+1\right)}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}% \mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac% {\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{% \nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),}}$
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DLMF:14.3.E16
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${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$
C5.S2.E1.m2aidec
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${\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}$
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${\displaystyle{\displaystyle\pi}}$
C3.S12.E1.m2aedec
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${\displaystyle{\displaystyle\cos\NVar{z}}}$
C4.S14.E2.m2acdec
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${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}$
C15.S2.E2.m2aldec
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${\displaystyle{\displaystyle x}}$
C14.S1.XMD1.m1odec
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${\displaystyle{\displaystyle\mu}}$
C14.S1.XMD7.m1kdec
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${\displaystyle{\displaystyle\nu}}$
C14.S1.XMD8.m1ndec
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