Entry 150

Ramanujan's theory of elliptic functions to alternative bases considers the hypergeometric function \(_2F_1(a,b;c;z)\) with \(a+b=c=1\) for the cases \(a=\frac16, \frac14, \frac13, \frac12\). We can relate this to the McKay-Thompson series \(j_n = j_n(\tau)\) for the Monster defined in Entry 145 for \(n = 1,2,3,4\). Define,

$$\alpha_1(\tau) = \frac12\left(1-\sqrt{1-\frac{1728}{j_{1}(\tau)}}\right) $$

where \(j = j_1(\tau)\) is the j-function. Let \(\alpha_1 = \alpha_1(\tau)\). Then we conjecture that,

$$\frac{_2F_1\big(\frac16,\frac56,1,\,1-\alpha_1\big)}{_2F_1\big(\frac16,\frac56,1,\,\alpha_1\big)}=-\tau\sqrt{-1}$$ as well as

$$\begin{align}j_{1}(\tau)  &=\frac{432}{\alpha_1\,(1-\alpha_1)}\\ &=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{8}+2^8 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{16}\right)^3\\  &=\left(\frac{_2F_1\big(\frac16,\frac56,1,\,\alpha_1\big)}{\eta^2(\tau)} \right)^{24/2}\end{align}$$

Example. Let \(\tau =\sqrt{-3}\). Then \(\alpha_1=\frac1{5\sqrt5}\left(\frac{-1+\sqrt5}2\right)^5\) solves $$\frac{_2F_1\big(\frac16,\frac56,1,\,1-\alpha_1\big)}{_2F_1\big(\frac16,\frac56,1,\,\alpha_1\big)}=\sqrt3$$ since \(-\tau\sqrt{-1}=-\sqrt3\, i\times i=\sqrt3\).