Entry 127

Assume \(\tau=n\sqrt{-\color{blue}{3}}\) for some positive integer \(n\). Given \(_2F_1(a,b;c;z)\) where \(a+b=c=\frac12\) for the case \(a=\tfrac1{6}\). Let \(z_1 = (1-2w)^2\) where \(w\) is $$w=\frac{27}{27+\Big(\tfrac{\eta(\tau/3)}{\eta(\tau)}\Big)^{12}}$$ Then \((z_1, z_2)\) are algebraic numbers in

$$_2F_1\left(\tfrac16,\tfrac13;\tfrac12;z_1\right) = z_2$$

Example: 

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

$$_2F_1\left(\frac16,\frac13;\frac12;\,\frac{25}{27}\right)=\frac{3\sqrt3}{4}$$ 

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

$$\qquad _2F_1\left(\frac16,\frac13;\frac12;\,\frac{45^2\,(\sqrt2+\sqrt3)^2}{(1+\sqrt6)^8}\right)=\frac58\big(1+\sqrt6\big)$$