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Journal of the London Mathematical Society Advance Access originally published online on January 31, 2007
Journal of the London Mathematical Society 2007 75(1):225-242; doi:10.1112/jlms/jdl017
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© 2007 London Mathematical Society

Ramanujan's Eisenstein series and powers of Dedekind's eta-function

Heng Huat Chan

National University of Singapore
Department of Mathematics
2 Science Drive 2
Singapore 117543
matchh{at}nus.edu.sg

Shaun Cooper

Massey University
Albany Campus
Private Bag 102 904
North Shore Mail Centre
Auckland
New Zealand
s.cooper{at}massey.ac.nz

Pee Choon Toh

National University of Singapore
Department of Mathematics
2 Science Drive 2
Singapore 117543
peechoon{at}nus.edu.sg

In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald's identities for A2, B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form {eta}d({tau})F(P, Q, R), where {eta}({tau}) is Dedekind's eta-function, and F(P, Q, R) is a polynomial in Ramanujan's Eisenstein series P, Q and R. We also derive identities in the case when d = 26. These lead to a new expression for {eta}26({tau}). This work generalizes the results for d = 1 and d = 3 which were given by Ramanujan on page 369 of ‘The Lost Notebook’.

2000 Mathematics Subject Classification 11F11, 11F20, 14K25, 33D52, 33E05.

Received January 23, 2006; published online January 31, 2007.


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