Some Eisenstein series identities

In this paper we use the theory of elliptic functions to provide different proofs of some Eisenstein series identities of Ramanujan from those given i n a recent paper by B. C. Berndt, S. Bhargava, and F. G. Garvan (1995, Tran s. Amer. Math. Soc. 347, 4136-4244). From one of these identities we derive the inversion formula for the Borweins cubic theta functions via Venkatach anliengar's method. We also derive some striking Eisenstein series identiti es associated with the Borweins' cubic theta functions. (C) 2000 Academic P ress.