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

Abstract

In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald's identities for A 2, B2 and G2. Using these identities, we express, for d = 8,10 or 14, certain theta functions in the form η (τ)F(P, Q, R), where η(τ) 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 η26(τ). This work generalizes the results for d = 1 and d = 3 which were given by Ramanujan on page 369 of 'The Lost Notebook'. © 2007 London Mathematical Society.