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F. Garvan;
Cubic modular identities of Ramanujan
,
Trans Amer. Math. Soc, 343 (1994), 35-47.
Abstract:
There is a beautiful cubic analogue of Jacobi's fundamental theta
function identity: $\theta_3^4 = \theta_4^4 + \theta_2^4$.
(see paper for explicit form)
In this note we provide an elementary
proof of this identity and of a related identity due to Ramanujan.