Entry 187

For the fourth McKay-Thompson series for the Monster $$j_{4}(\tau) = \left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{4} + 4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)^2 = \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \right)^{24}$$Since we have Ramanujan's \(G_d\)-function for \(\tau = \sqrt{-d}\) as

$$2^{1/4}G_d = \frac{\eta^2(\tau)}{\eta(\tfrac12\tau)\,\eta(2\tau)}$$

then \(j_4\big(\tfrac12\tau\big)\) is just the \(24\)th power of \(2^{1/4}G_d\). Hence we propose 

$$K(k_d) = \frac{\pi}2 \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac1{\big(2^{1/4}G_d\big)^{24n}}}$$

Examples. The values for \(G_5, G_{13}, G_{37}\) have already been given in Entry 181 and involve roots of quadratics, like \(G_5 = \phi^{1/4}\) with golden ratio \(\phi\). But for cubics, given the tribonacci constant \(T\), the plastic ratio \(P\) and the supergolden ratio \(\psi\), the real root of

$$\begin{align}T^3-T^2-T-1 &= 0\\ P^3-P-1 &=0\\ \psi^3-\psi^2-1 &=0 \end{align}$$

then \(G_{11}, G_{23}, G_{31}\) are

$$\begin{align}G_{11} &= 2^{-1/4}\left(\tfrac{T+1}T\right) \\ G_{23} &= 2^{1/4}P \\ \,G_{31} &=2^{1/4}\psi\end{align}$$

and we conjecture

$$\quad K(k_5) = \frac{\pi}2 \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac1{\big(2^{1/4}\phi^{1/4}\big)^{24n}}}$$

$$K(k_{11}) = \frac{\pi}2 \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac1{\big(\tfrac{T+1}T\big)^{24n}}}\quad$$

$$K(k_{23}) = \frac{\pi}2 \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac1{\big(2^{1/2}P\big)^{24n}}}$$

$$K(k_{31}) = \frac{\pi}2 \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac1{\big(2^{1/2}\psi\big)^{24n}}}$$

and so on.