Entry 132

Given $_2F_1(a,b;c;z)$ and Dedekind eta function $\eta(\tau)$ where $\tau = \frac{1+n\sqrt{-3}}2$ for positive integer $n$. Then for type $a+b=c=\color{blue}{\tfrac56}$ $$\begin{align}&\,_2F_1\big(\tfrac12,\tfrac13;\tfrac56;(1-2\delta_1)^2\big),\quad\;\delta_1 = \delta_2\\ &\,_2F_1\big(\tfrac13,\tfrac12;\tfrac56;(1-2\delta_2)^2\big),\quad\;\frac1{\delta_2}-1=\sqrt{\frac1{27}\left(\tfrac{\eta\big(\frac{\tau+1}{3}\big)}{\eta(\tau)}\right)^{12}}\\ &\,_2F_1\big(\tfrac14,\tfrac7{12};\tfrac56;(1-2\delta_3)^2\big),\quad\color{red}{\delta_3 =\,?} \\ &\,_2F_1\big(\tfrac16,\tfrac23;\tfrac56;(1-2\delta_4)^2\big),\quad\;\frac1{\delta_4}-1\,=\,\frac1{27}\left(\tfrac{\eta\big(\frac{\tau+1}{3}\big)}{\eta(\tau)}\right)^{12}\end{align}$$ For this type, there are infinitely many hypergeometrics such that both $(z_1, z_2)$ in $$_2F_1(a,b;c;z_1) = z_2$$ are algebraic numbers when $n$ is a positive integer. Note that $_2F_1\big(\tfrac12,\tfrac13;\tfrac56;z\big) =\,_2F_1\big(\tfrac13,\tfrac12;\tfrac56;z\big)$ so the first form is superfluous. Examples: Let $\tau = \frac{1+5\sqrt{-3}}2$, $$_2F_1\Big(\frac13,\frac12;\frac56;\frac45\Big)=\;\frac35\sqrt5$$ $$\quad _2F_1\Big(\frac16,\frac23;\frac56;\frac{80}{81}\Big)=\frac35\,(9\sqrt5)^{1/3}$$