Entry 128

Assume \(\tau=n\sqrt{-\color{blue}{4}}\) for some positive integer \(n\). Given \(_2F_1(a,b;c;z)\) where \(a+b=c=\frac12\) for the case \(a=\tfrac1{4}\). Let \(z_1 = (1-2w)^2\) where \(w\) is $$w=\frac{16}{16+\Big(\tfrac{\eta(\tau/4)}{\eta(\tau)}\Big)^8}$$ Then \((z_1, z_2)\) are algebraic numbers in

$$_2F_1\left(\tfrac14,\tfrac14;\tfrac12;z_1\right) = z_2$$

Examples: 

If \(n=2\) so \(\tau=2\sqrt{-4}\), then,

$$_2F_1\left(\frac14,\frac14;\frac12;\,\frac{9\,(3+\sqrt2)^2}{(1+\sqrt2)^6}\right)=\frac38\big(2+\sqrt2\big)$$ 

If \(n=3\) so \(\tau=3\sqrt{-4}\), then,

$$\quad _2F_1\left(\frac14,\frac14;\frac12;\,\frac{8\sqrt3}{(2+\sqrt3)^2}\right)=\frac23\big(3+2\sqrt3\big)^{1/2}$$