I would like to ask a question about hypergeometric function ${}_2F_1$. Specifically, I hope to solve the equation $$ \frac{\delta x}{1-\delta}{}_2F_1\left(1,1-\delta;2-\delta;x\right)=a, $$ where $\delta\in\left(0,1\right)$ and $a\geq 1$. Is the $x$ has the closed-form solution? If not, for the special case when $\delta=\frac{1}{2}$, can we get $x$?
I think the above hypergeometric function has the special structure, but I checked the literatures and cannot find useful results. I appreciate if you can help to solve it, or give some suggestions.
Thanks.