Thompson Series and Ramanujan’s Identities

  • Masao Koike
Part of the Developments in Mathematics book series (DEVM, volume 11)

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

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References

  1. [1]
    B. J. Birch, A look back at Ramanujan’s Notebooks, Math. Proc. Camb. Phil. Soc. 78 (1975), 73–79.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. J. F. Biagioli, A proof of some identities of Ramanujan using modular forms, Glasgow Math. J. 31 (1989), 271–295.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), 308–309.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J. Mckay and H. StraussThe q-series of monstrous moonshine and the decomposition of the head characters, Comm. in Algebra 18 (1990), 253–278.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Masao Koike
    • 1
  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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