I need to proof Pfaff's law:
It's supposed to be proven 'by comparing the coefficients of $\displaystyle z^n$ on both sides of the equation'.
I don't know much about hypergeometric functions, but according to this article (pdf file), $\displaystyle F\left({a,b\atop c}\,\Big|\,z\right)$ is given by the integral
$\displaystyle F\left({a,b\atop c}\,\Big|\,z\right) = \frac1{\mathrm{B}(b,c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt,$
where the B denotes the beta function.
If you are willing to accept that formula, then the relation $\displaystyle F\left({a,c-b\atop c}\,\Big|\,z\right) = \frac1{(1-z)^a}F\left({a,b\atop c}\,\Big|\,\frac{-z}{1-z}\right)$ follows very easily by making the substitution $\displaystyle t\mapsto1-t$ in the integral.