Equation (9.6.8) in Lebedev (Special Functions and Their Applications) reads
$F\left(\alpha ,\beta ;\frac{1}{2} (\alpha +\beta +1);z\right)=F\left(\frac{\alpha }{2},\frac{\beta }{2};\frac{1}{2} (\alpha +\beta +1);4 z (1-z)\right)$
with the restrictions $\Re(z)<\frac{1}{2}$ and $\frac{1}{2} (\alpha +\beta +1)\neq0,-1,-2,...$
I understand the second restriction, but cannot see where the first comes from. The derivation is based on an earlier formula (9.6.1) where we have the usual restriction $| \arg (1-z)| <\pi$ and make the substitution $\frac{1}{2} \left(1-\sqrt{1-z}\right)\to z$
This maps the original branch cut [1,$\infty$) to [1/2, 1/2-i $\infty$)