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 A2, B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form ηd(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’.