Entry 153

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,

$$\alpha_4(\tau) = \frac{16}{\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{8}+16} = \left(\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}\right)^8$$

Let \(\alpha_4 = \alpha_4(\tau)\). Then we conjecture that,

$$\frac{_2F_1\big(\frac12,\frac12,1,\,1-\alpha_4\big)}{_2F_1\big(\frac12,\frac12,1,\,\alpha_4\big)}=-\tau\sqrt{-4}$$ as well as

$$\begin{align}j_{4}(\tau) &=  \frac{16}{\alpha_4\,(1-\alpha_4)}\\ &= \left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{4}+4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)^2 = \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \right)^{24}\\ &= \left(\frac{_2F_1\big(\frac12,\frac12,1,\,\alpha_4\big)}{\eta^2(\tau)} \times\frac{\eta(\tau)}{\eta(4\tau)}\right)^{24/5}\end{align}$$

Example. Let \(\tau =\sqrt{-3}\). Then \(\alpha_4=(\sqrt3-\sqrt2)^4(\sqrt2-1)^4\) solves $$\frac{_2F_1\big(\frac12,\frac12,1,\,1-\alpha_4\big)}{_2F_1\big(\frac12,\frac12,1,\,\alpha_4\big)}=\sqrt4\times\sqrt3$$ Alternatively and more familiar

$$\frac{_2F_1\big(\frac12,\frac12,1,\,1-\lambda(\sqrt{-r})\big)}{_2F_1\big(\frac12,\frac12,1,\,\lambda(\sqrt{-r})\big)}=\sqrt{r}$$ where \(\lambda(\tau)\) is the modular lambda function.