\[\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c}%
;\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}%
\right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right),\]