On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions

One of the many amazing things Ramanujan did in his lifetime was to list $40$ identities involving what are now called the Rogers-Ramanujan functions $G(q)$ and $H(q)$ on one side, and products of functions of the form $Q_m = \prod_{n=1}^\infty (1-q^{mn})$ on the other side. The identities are rather complicated and seem too difficult to guess. Recently however, Koike devised a strategy for finding (but not proving) these types of identities by connecting them to Thompson series. He was able to conjecture many new Rogers-Ramanujan type identities between $G(q)$ and $H(q)$, and Thompson series. Here we prove these identities.