# DLMF:15.8.E5 (Q5062)

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DLMF:15.8.E5
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## Statements

${\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}% \left({a,b\atop c};z\right)=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-% b\right)}\mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-% z)^{c-a-b}z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c% -a,1-a\atop c-a-b+1};1-\frac{1}{z}\right),}}$
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DLMF:15.8.E5
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${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}$
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${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$
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${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$
C5.S2.E1.m2acdec
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${\displaystyle{\displaystyle\pi}}$
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${\displaystyle{\displaystyle\operatorname{ph}}}$
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${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}$
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${\displaystyle{\displaystyle\sin\NVar{z}}}$
C4.S14.E1.m2acdec
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${\displaystyle{\displaystyle z}}$
C15.S1.XMD3.m1ddec
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${\displaystyle{\displaystyle a}}$
C15.S1.XMD4.m1ddec
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${\displaystyle{\displaystyle b}}$
C15.S1.XMD5.m1ddec
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${\displaystyle{\displaystyle c}}$
C15.S1.XMD6.m1ddec
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