Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some cubic modular identities of Ramanujan


Authors: J. M. Borwein, P. B. Borwein and F. G. Garvan
Journal: Trans. Amer. Math. Soc. 343 (1994), 35-47
MSC: Primary 11B65; Secondary 11F27, 33D10
MathSciNet review: 1243610
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle {\left({\sum\limits_{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m... ...+ (n + \frac{1}{3})(m + \frac{1}{3}) + {{(m + \frac{1}{3})}^2}}}} } \right)^3}.$

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11B65, 11F27, 33D10

Retrieve articles in all journals with MSC: 11B65, 11F27, 33D10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1243610-6
PII: S 0002-9947(1994)1243610-6
Keywords: Theta functions, q-series, eta function, modular forms, cubic modular equations, hypergeometric functions
Article copyright: © Copyright 1994 American Mathematical Society