Borwein, Jonathan M. and Borwein, Peter and Garvan, F.G. (1994) *Some cubic modular identities of Ramanujan.* Trans. Amer. Math. Soc., 343 . pp. 35-48.

## Abstract

There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is $(\sum_{n,m=-\infty}^{\infty} q^{n^2+nm+m^2})³ = (\sum_{n,m=-\infty}^{\infty} ω^{n-m}q^{n²+nm+m²})³ + (\sum_{n,m=-\infty}^{\infty} q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³.$ Here $ω = exp(2π i/3).$ In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.

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