Entry 131

Given \(_2F_1(a,b;c;z)\) and Dedekind eta function \(\eta(\tau)\) where \(\tau = \frac{1+n\sqrt{-1}}2\) for positive integer \(n\). Then for type \(a+b=c=\color{blue}{\tfrac34}\) $$\begin{align}&\,_2F_1\big(\tfrac14,\tfrac12;\tfrac34;(1-2\gamma_1)^2\big), \quad\frac1{\gamma_1}-1=\sqrt{-\frac1{64}\Big(\tfrac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}\Big)^{24}}\\ &\,_2F_1\big(\tfrac16,\tfrac7{12};\tfrac34;(1-2\gamma_2)^2\big),\quad\color{red}{\gamma_2 =\,?} \\ &\,_2F_1\big(\tfrac18,\tfrac58;\tfrac34;(1-2\gamma_3)^2\big),\quad\frac1{\gamma_3}-1\,=\, -\frac1{64}\Big(\tfrac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}\Big)^{24}\\ &\,_2F_1\big(\tfrac1{12},\tfrac23;\tfrac34;(1-2\gamma_4)^2\big),\quad\color{red}{\gamma_4 =\,?}\end{align}$$ 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. Examples: Let \(\tau = \frac{1+5\sqrt{-1}}2\), 

$$_2F_1\Big(\frac14,\frac12;\frac34;\frac{80}{81}\Big)=\frac95$$

$$\quad _2F_1\Big(\frac18,\frac58;\frac34;\frac{25920}{25921}\Big)=\frac35\,161^{1/4}$$