# Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here.

In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London Mathematical Society, 2, XVIII, 1920) Ramanujan mentioned about the functions $G(q)$ and $H(q)$ which are part of Rogers-Ramanujan Identities: \begin{align}G(q) &= 1 + \frac{q}{1 - q} + \frac{q^{4}}{(1 - q)(1 - q^{2})} + \frac{q^{9}}{(1 - q)(1 - q^{2})(1 - q^{3})} + \cdots\\ &= \frac{1}{(1 - q)(1 - q^{6})(1 - q^{11})\cdots}\times\frac{1}{(1 - q^{4})(1 - q^{9})(1 - q^{14})\cdots}\tag{1}\end{align} \begin{align}H(q) &= 1 + \frac{q^{2}}{1 - q} + \frac{q^{6}}{(1 - q)(1 - q^{2})} + \frac{q^{12}}{(1 - q)(1 - q^{2})(1 - q^{3})} + \cdots\\ &= \frac{1}{(1 - q^{2})(1 - q^{7})(1 - q^{12})\cdots}\times\frac{1}{(1 - q^{3})(1 - q^{8})(1 - q^{13})\cdots}\tag{2}\end{align} Ramanujan gave a proof of these identities in another paper (which is described in this blog post).

Ramanujan mentions further "I have now found an algebraic relation between $G(q)$ and $H(q)$, viz.": $$H(q)\{G(q)\}^{11} - q^{2}G(q)\{H(q)\}^{11} = 1 + 11q\{G(q)H(q)\}^{6}\tag{3}$$ which is equivalent to $$\frac{1}{R^{5}(q)} - 11 - R^{5}(q) = \frac{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{6}}{q\{(1 - q^{5})(1 - q^{10})(1 - q^{15})\cdots\}^{6}}\tag{4}$$ where $R(q) = q^{1/5}H(q)/G(q)$ and is proved here.

In his characteristic style he ends his paper with another formula without proof. To quote him: "Another noteworthy formula is $$H(q)G(q^{11}) - q^{2}G(q)H(q^{11}) = 1\tag{5}$$ Each of these formulae is the simplest of a large class."

Is there any proof available for the formula $(5)$? Are the formulas belonging to "a large class" as envisaged by Ramanujan discovered and proved in literature? Any references or a proof would be highly appreciated.