Thompson Series and Ramanujan’s Identities

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Abstract

B. J. Birch published some manuscripts of Ramanujan which contained, among other things, a list of identities involving Rogers-Ramanujan functions $$ G(x) = \prod\limits_{n = 0}^\infty {\frac{1}{{(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})}}} $$ $$ H(x) = \prod\limits_{n = 0}^\infty {\frac{1}{{(1 - {x^{5n + 2}})(1 - {x^{5n + 3}})}}} $$ and $$ q(m) = \prod\limits_{n = 1} {(1 - {x^{mn}})} $$ which is related to Dedekind η functions $$ \eta (T) = {e^{\frac{{2\pi iT}}{{24}}}}\prod\limits_{n = 1}^\infty {(1 - {e^{2n\pi iT}})} $$ .