What is the effect of the Möbius transformation $z \rightarrow \frac{z}{z-1}$ on the points $z=0$, $z=\infty$ and $z=1$ ?

By considering

$$(z-1)^{-a} P\left\{\begin{array}{cccc} 0 & \infty & 1 & \\ 0 & a & 0 & z(z-1)^{-1} \\ 1-c & c-b & b-a & \end{array}\right\}$$

or otherwise, show that $(z-1)^{-a} F\left(a, c-b ; c ; z(z-1)^{-1}\right)$ is a branch of the $P$-function

Give a linearly independent branch.