I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$.

Here is an example called the Kummer quadratic transformation, which may be known to most of you. But it involves $x$ and $4x(1-x)$.

http://archive.lib.msu.edu/crcmath/math/math/k/k154.htm

Thanks.