Entry 151

Ramanujan's theory of elliptic functions to alternative bases can be related to the McKay-Thompson series \(j_n = j_n(\tau)\) for the Monster defined in Entry 145. Define,

$$\alpha_2(\tau) = \frac{64}{\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}+64} = \left(\frac{8}{\left(\frac{\eta(\tau/2)}{\eta(2\tau)}\right)^{8}+8}\right)^2 $$

Let \(\alpha_2 = \alpha_2(\tau)\). Then we conjecture that,

$$\frac{_2F_1\big(\frac14,\frac34,1,\,1-\alpha_2\big)}{_2F_1\big(\frac14,\frac34,1,\,\alpha_2\big)}=-\tau\sqrt{-2}$$ as well as

$$\begin{align}j_{2}(\tau) &=  \frac{64}{\alpha_2\,(1-\alpha_2)}\\ &= \left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{12}+2^6 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{12}\right)^2\\ &= \left(\frac{_2F_1\big(\frac14,\frac34,1,\,\alpha_2\big)}{\eta^2(\tau)} \times\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24/3}\end{align}$$

Example. Let \(\tau =\sqrt{-3}\). Then \(\alpha_2=\frac1{3(2+\sqrt3)^2(13+4\sqrt3)}\) solves $$\frac{_2F_1\big(\frac14,\frac34,1,\,1-\alpha_2\big)}{_2F_1\big(\frac14,\frac34,1,\,\alpha_2\big)}=\sqrt2\times\sqrt3$$