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 Qm = Pi(infinity)(n= 1) (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.