Entry 126

Assume \(\tau=n\sqrt{-\color{blue}{2}}\) for some positive integer \(n\). Given \(_2F_1(a,b;c;z)\) where \(a+b=c=\frac12\) for the case \(a=\tfrac1{8}\). Let \(z_1 = (1-2w)^2\) where \(w\) is $$w=\frac{64}{64+\Big(\tfrac{\eta(\tau/2)}{\eta(\tau)}\Big)^{24}}$$ Then \((z_1, z_2)\) are algebraic numbers in

$$_2F_1\left(\tfrac18,\tfrac38;\tfrac12;z_1\right) = z_2$$

Examples: 

If \(n=2\) so \(\tau=2\sqrt{-2}\), then,

$$_2F_1\left(\frac18,\frac38;\frac12;\frac{(70\sqrt{2})^2\,(1+\sqrt2)^2}{(2+3\sqrt2)^6}\right)=\frac34\left(\frac{2+3\sqrt2}2\right)^{1/2}$$

If \(n=3\) so \(\tau=3\sqrt{-2}\), then,

$$_2F_1\left(\frac18,\frac38;\frac12;\frac{2400}{2401}\right)=\tfrac23\cdot7^{1/2}$$