DLMF:15.8.E11 (Q5068)

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DLMF:15.8.E11
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${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^% {-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma% \left(b+m-k\right)}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a% \right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(b-k\right)}(-1% )^{k}\left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-% \psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b% -k\right)\right),}}$
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DLMF:15.8.E11
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${\displaystyle{\displaystyle\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|% \operatorname{ph}\left(1-z\right)|<\pi}}$
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${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$
C5.S2.E1.m2agdec
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${\displaystyle{\displaystyle\pi}}$
C3.S12.E1.m2ahdec
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${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$
C5.S2.E2.m2acdec
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${\displaystyle{\displaystyle!}}$
introduction.Sx4.p1.t1.r15.m5acdec
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${\displaystyle{\displaystyle\ln\NVar{z}}}$
C4.S2.E2.m2acdec
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${\displaystyle{\displaystyle\operatorname{ph}}}$
C1.S9.E7.m1ahdec
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${\displaystyle{\displaystyle\Re}}$
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${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}$
C15.S2.E2.m2ahdec
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${\displaystyle{\displaystyle z}}$
C15.S1.XMD3.m1jdec
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${\displaystyle{\displaystyle a}}$
C15.S1.XMD4.m1hdec
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${\displaystyle{\displaystyle b}}$
C15.S1.XMD5.m1hdec
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${\displaystyle{\displaystyle k}}$
C15.S1.XMD7.m1cdec
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${\displaystyle{\displaystyle m}}$
C15.S1.XMD9.m1edec
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