Generalized Jacobi Theta functions, Macdonald's identities and powers of Dedekind's ETA function

Abstract

Ramanujan (1919) studied expansions of the form n^d(t)E(t) for d=1 or 3 where n(t) is Dedekind's eta function and E(t) is some polynomial generated by Ramanujan's Eisenstein series. In another direction, Newman (1955) and Serre (1985) used the theory of modular forms to prove that n^d(t) is lacunary and is annihilated by certain Hecke operators whenever d=2, 4, 6, 8, 10, 14 and 26.In this thesis, we generalize the results of Ramanujan, Newman and Serre by constructing infinitely many expansions for n^d(t)E(t) where d=2, 4, 6, 8, 10, 14 and 26. In particular, the last four cases are new. We use invariance properties of generalized Jacobi theta functions to construct identities equivalent to the Macdonald identities for A_2, B_2 and G_2, to establish the cases d=8, 10 and 14. The case for d=26 is proved using the theory of modular forms. By restricting to modular forms, we also obtain infinitely many expansions of n^d(t)F(t) where d=2, 4, 6, 8, 10, 14, 26 and F(t) is a modular form. We then continue to use generalized Jacobi theta functions to construct determinant identities equivalent to the Macdonald identities for all the infinite families and use them to deduce new formulas for higher powers of n^d(t).