Chapter

Galois Theory and Modular Forms

Volume 11 of the series Developments in Mathematics pp 367-373

Thompson Series and Ramanujan’s Identities

  • Masao KoikeAffiliated withGraduate School of Mathematics, Kyushu University

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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
$$ \eta (T) = {e^{\frac{{2\pi iT}}{{24}}}}\prod\limits_{n = 1}^\infty {(1 - {e^{2n\pi iT}})} $$
.