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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The root lattice $A^*_n$ and Ramanujan's circular summation of theta functions


Author: Kok Seng Chua
Journal: Proc. Amer. Math. Soc. 130 (2002), 1-8
MSC (2000): Primary 11B65, 11E20
Published electronically: May 3, 2001
MathSciNet review: 1855612
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Abstract:

We relate a formula of Ramanujan on the circular summation of the $n$th power of theta functions, $F_n(q)$, to the theta series of the root lattice $A^*_n$. We then use properties of the lattice to show that $F_n$ includes an $\operatorname{SL}_2(\mathbf{Z})$ modular form when $n$ is an odd perfect square as well as to derive a very simple expression for $F_9(q)$.


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Additional Information

Kok Seng Chua
Affiliation: Institute of High Performance Computing, 89C Science Park Driver, #02-11/12 The Rutherford, Singapore 118261
Email: chuaks@ihpc.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06080-4
PII: S 0002-9939(01)06080-4
Keywords: Ramanujan, theta functions, lattices
Received by editor(s): May 12, 2000
Published electronically: May 3, 2001
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2001 American Mathematical Society