## Some cubic modular identities of Ramanujan

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- 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
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## 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.## References

- George E. Andrews,
*The theory of partitions*, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR**0557013** - Richard Bellman,
*A brief introduction to theta functions*, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR**0125252**, DOI 10.1017/s0025557200044491 - Bruce C. Berndt,
*Ramanujan’s notebooks. Part III*, Springer-Verlag, New York, 1991. MR**1117903**, DOI 10.1007/978-1-4612-0965-2 - Jonathan M. Borwein and Peter B. Borwein,
*Pi and the AGM*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. MR**877728** - J. M. Borwein and P. B. Borwein,
*A cubic counterpart of Jacobi’s identity and the AGM*, Trans. Amer. Math. Soc.**323**(1991), no. 2, 691–701. MR**1010408**, DOI 10.1090/S0002-9947-1991-1010408-0
—, - John A. Ewell,
*On the enumerator for sums of three squares*, Fibonacci Quart.**24**(1986), no. 2, 150–153. MR**843964** - Nathan J. Fine,
*Basic hypergeometric series and applications*, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988. With a foreword by George E. Andrews. MR**956465**, DOI 10.1090/surv/027
O. Kolberg, - Louis W. Kolitsch,
*A congruence for generalized Frobenius partitions with $3$ colors modulo powers of $3$*, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 343–348. MR**1084189**
L. Lorenz, - Srinivasa Ramanujan,
*Notebooks. Vols. 1, 2*, Tata Institute of Fundamental Research, Bombay, 1957. MR**0099904** - Srinivasa Ramanujan,
*The lost notebook and other unpublished papers*, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR**947735** - Bruno Schoeneberg,
*Elliptic modular functions: an introduction*, Die Grundlehren der mathematischen Wissenschaften, Band 203, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German by J. R. Smart and E. A. Schwandt. MR**0412107**

*A remarkable cubic iteration*, Computational Methods and Function Theory, Lecture Notes in Math., vol. 1435, Springer-Verlag, New York, 1990.

*Note on the Eisenstein series of*${\Gamma _0}(p)$, Universitet i Bergen Årbok, Naturvitenskapelig rekke, Nr. 15, 1959.

*Bidrag til tallenes theori*, Tidsskrift for Mathematik (3)

**1**(1871), 97-114.

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- 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