Entry 188

Ramanujan g-function \(g_m\) and \(\tau = \sqrt{-m}\) is,

$$2^{1/4}g_m = \frac{\eta\big(\frac12\tau\big)}{\eta(\tau)}$$ Ramanujan tabulated \(G_m\) and \(g_m\) for odd and even \(m\), respectively. For the latter and \(m\geq4\), we propose a slightly different formula,

$$\,_2F_1\big(\tfrac12,\tfrac12;1,\lambda(1+\sqrt{-m})\big) = \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac{(-1)^n}{\big(2^{1/4}g_m\big)^n}}$$

with modular lambda function \(\lambda(z)\). Values for \(g_m\) have been given in Entry 183. Using some of them, we conjecture

$$\,_2F_1\big(\tfrac12,\tfrac12;1,\lambda(1+\sqrt{-6})\big) = \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac{(-1)^n}{\big(2^{1/4}(1+\sqrt2)^{1/6}\big)^n}}$$

$$\,_2F_1\big(\tfrac12,\tfrac12;1,\lambda(1+\sqrt{-22})\big) = \sqrt{\sum_{n=0}^\infty \frac{(2n)!^3}{n!^6} \frac{(-1)^n}{\big(2^{1/4}(1+\sqrt2)^{1/2}\big)^n}}$$

and so on.