# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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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
DOI: https://doi.org/10.1090/S0002-9947-1994-1243610-6

## 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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Bibliographic Information
• Journal: Trans. Amer. Math. Soc. 343 (1994), 35-47
• MSC: Primary 11B65; Secondary 11F27, 33D10
• DOI: https://doi.org/10.1090/S0002-9947-1994-1243610-6
• MathSciNet review: 1243610