Entry 154

As an overview of the previous four entries, Ramanujan's theory of elliptic functions to alternative bases uses the hypergeometric function \(_2F_1(a,b;c;z)\) with \(a+b=c=1\) for the cases \(a=\frac16, \frac14, \frac13, \frac12\). This can be related to the McKay-Thompson series \(j_n = j_n(\tau)\) for the Monster defined in Entry 145 for \(n = 1,2,3,4\). Consider the equations, 

$$\begin{align}\frac{_2F_1\big(\frac16,\frac56,1,\,1-\alpha_1\big)}{_2F_1\big(\frac16,\frac56,1,\,\alpha_1\big)} &=-\tau\sqrt{-1}\\ \frac{_2F_1\big(\frac14,\frac34,1,\,1-\alpha_2\big)}{_2F_1\big(\frac14,\frac34,1,\,\alpha_2\big)} &=-\tau\sqrt{-2}\\ \frac{_2F_1\big(\frac13,\frac23,1,\,1-\alpha_3\big)}{_2F_1\big(\frac13,\frac23,1,\,\alpha_3\big)} &=-\tau\sqrt{-3}\\ \frac{_2F_1\big(\frac12,\frac12,1,\,1-\alpha_4\big)}{_2F_1\big(\frac12,\frac12,1,\,\alpha_4\big)} &=-\tau\sqrt{-4}\end{align}$$ Then the \(\alpha_n\) can be solved and expressed in terms of the \(j_n(\tau)\) as discussed in the said entries.