# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Some cubic modular identities of RamanujanHTML articles powered by AMS MathViewer

by J. M. Borwein, P. B. Borwein and F. G. Garvan
Trans. Amer. Math. Soc. 343 (1994), 35-47 Request permission

## Abstract:

There is a beautiful cubic analogue of Jacobi’s fundamental theta function identity: $\theta _3^4 = \theta _4^4 + \theta _2^4$. It is ${\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m^2}}}} } \right )^3} = {\left ({\sum \limits _{n,m = - \infty }^\infty {{\omega ^{n - m}}{q^{{n^2} + nm + {m^2}}}} } \right )^3} + {\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{{(n + \frac {1}{3})}^2} + (n + \frac {1}{3})(m + \frac {1}{3}) + {{(m + \frac {1}{3})}^2}}}} } \right )^3}.$ Here $\omega = \exp (2\pi i/3)$. 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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