In fact your function is elementary and very simple, for its explicit form look at Brychkov,Marichev,Prudnikov, Integral and Series, vol.3 : $$ F(a,a+1/2;2a;z)=\frac{1}{\sqrt{1-z}}\left(\frac{2}{1+\sqrt{1-z}}\right)^{2a-1}. $$
Unfortunately I can not give a reference to the exact formula, this vol. 3 is the only one I have not an English version.
Also sorry for may be somehow akward post -- it was my first answer here, be patient.