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

Abstract

One of the many amazing things Ramanujan did in his lifetime was to list 40 40 identities involving what are now called the Rogers-Ramanujan functions G ( q ) G(q) and H ( q ) H(q) on one side, and products of functions of the form Q m = ∏ n = 1 ∞ ( 1 − q m n ) 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 ) G(q) and H ( q ) H(q) , and Thompson series. Here we prove these identities.