Entry 179

For the second McKay-Thompson series of the Monster, or \(j_2(\tau)\) 

$$j_2(\tau) =\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{12}+2^6\left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{12}\right)^2$$

we conjecture that the complete elliptic integral of the first kind \(K(k_d)\)

$$K(k_d) = \frac{\pi}2  \frac{\beta^3(\tau)}{\;\big({-j_2(\tau)}\big)^{1/8}}\, \sqrt{\sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac1{\big(j_2(\tau)\big)^n}}$$

$$K(k_d) = \frac{\pi}2  \frac{(2^{1/4}G_d)^3}{\;\big({-j_2(\tau)}\big)^{1/8}}\, \sqrt{\sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac1{\big(j_2(\tau)\big)^n}}$$

with Ramanujan G-function \(G_d\) for integer \(d\geq5\) and \(\tau = \frac{1+\sqrt{-d}}2\) since, 

$$\beta(\tau) = 2^{1/4}G_d = \frac{\eta^2(\sqrt{-d})}{\eta(\tfrac12\sqrt{-d})\,\eta(2\sqrt{-d})} = \zeta_{48}\frac{\eta(\tau)}{\eta(2\tau)}$$and \(48\)th root of unity \(\zeta_{48} = e^{2\pi i/48}\). 

Examples. For class number \(h(-d) = 2\), it is known that 

$$\begin{align}G_{5} &= \left(\frac{1+\sqrt{5}}2\right)^{1/4}\\ G_{13} &= \left(\frac{3+\sqrt{13}}2\right)^{1/4}\\ \,G_{37} &=\, \big(6+\sqrt{37}\big)^{1/4}\end{align}$$

hence,

$$K(k_{5}) = \frac{\pi}2  \frac{\big(1+\sqrt{5}\big)^{3/4}}{\big(4\sqrt2 \big)^{1/2}}\, \sqrt{\sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac1{\big({-(4\sqrt2)^4}\big)^n}}$$

$$K(k_{13}) = \frac{\pi}2  \frac{\big(3+\sqrt{13}\big)^{3/4}}{\big(12\sqrt2 \big)^{1/2}}\, \sqrt{\sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac1{\big({-(12\sqrt2)^4}\big)^n}}$$

$$K(k_{37}) = \frac{\pi}2  \frac{2^{3/4}\big(6+\sqrt{37}\big)^{3/4}}{\big(84\sqrt2 \big)^{1/2}}\, \sqrt{\sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac1{\big({-(84\sqrt2)^4}\big)^n}}$$

and so on.