Entry 36

Given the nome \(p=e^{\pi i \tau}\) and \(\tau =\sqrt{-n}\), then the Ramanujan functions \(G_n\) and \(g_n\) are
$$\begin{aligned}G_n &= 2^{-1/4}p^{-1/24}\prod_{k=1,3,5,\dots}^\infty\big(1+p^k\big) \\ g_n &= 2^{-1/4}p^{-1/24}\prod_{k=1,3,5,\dots}^\infty\big(1-p^k\big)\end{aligned}$$ These are ubiquitous in Ramanujan's Notebooks. Well-known values are \(G_5 = \phi^{1/4}\) and \(G_{25} =\phi\), where \(\phi\) is the golden ratio. However, given the nome's square \(q=e^{2\pi i \tau}\), note also the three Weber modular functions,
$$\begin{aligned}\mathfrak{f}(\tau)&= q^{-1/48}\prod_{n=1}^\infty\big(1+q^{n-1/2}\big) \\ \mathfrak{f}_1(\tau)&= q^{-1/48}\prod_{n=1}^\infty\big(1-q^{n-1/2}\big) \\ \mathfrak{f}_2(\tau)&= \sqrt2\,q^{1/24}\prod_{n=1}^\infty\big(1+q^{n}\big) \end{aligned}$$ Notice the similarity of the definitions. Mathematica doesn't have built-in functions for these, but fortunately can be expressed by the more familiar Dedekind eta function \(\eta(\tau)\) with \(\tau =\sqrt{-n}\), $$G_n =2^{-1/4}\,\mathfrak{f}(\tau)=\frac{2^{-1/4}\,\eta^2(\tau)}{\eta\big(\tfrac{\tau}{2}\big)\,\eta(2\tau)}$$ $$g_n =2^{-1/4}\,\mathfrak{f}_1(\tau)= \frac{2^{-1/4}\,\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}$$The two functions \(G_n\) and \(g_n\) obey$$(G_n g_n)^8(G_n^8-g_n^8) = \tfrac{1}{4}$$which is consequence of Weber's$$\mathfrak{f}_1(\tau)^8+\mathfrak{f}_2(\tau)^8 = \mathfrak{f}(\tau)^8$$Ramanujan calculated many explicit values for \(G_n\) and \(g_n\), one of which is the remarkable \(G_{125}=3.6335\dots\)$$\frac{2\,G_{125}}{\varphi^{-5/4}}=-1+\frac{1}{\sqrt{5}}\left(1+2^{2/5}\varphi^{1/5}\left(\sqrt[5]{4-\sqrt{5}+(\varphi\sqrt{5})^{3/2}}+\sqrt[5]{4-\sqrt{5}-(\varphi\sqrt{5})^{3/2}}  \right)\right)^2$$ with the reciprocal golden ratio \(\varphi=\frac{-1+\sqrt{5}}{2}\approx0.61803\).