Rogers–Ramanujan continued fraction

Summary

The Rogers–Ramanujan continued fraction is $R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \cdots}}}$ for $\abs{q} < 1$. It equals $q^{1/5}$ times the quotient of the two series of the Rogers–Ramanujan identities, hence an infinite product. As a function of $\tau$ with $q = e^{2\pi i \tau}$ it is a modular function for the group $\Gamma(5)$. Ramanujan found closed forms of its values at special points, such as $R(e^{-2\pi}) = \sqrt{\frac{5 + \sqrt5}{2}} - \frac{1 + \sqrt5}{2}$.