The root lattice 𝐴*_{𝑛} and Ramanujan’s circular summation of theta functions

Chua, Kok

doi:10.1090/S0002-9939-01-06080-4

The root lattice $A^*_n$ and Ramanujan’s circular summation of theta functions
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by Kok Seng Chua

Proc. Amer. Math. Soc. 130 (2002), 1-8

DOI: https://doi.org/10.1090/S0002-9939-01-06080-4

Published electronically: May 3, 2001

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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 $F_9(q)$.

References

Scott Ahlgren, The sixth, eighth, ninth, and tenth powers of Ramanujan’s theta function, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1333–1338. MR 1646322, DOI 10.1090/S0002-9939-99-05181-3
Bruce C. Berndt, Ramanujan’s theory of theta-functions, Theta functions: from the classical to the modern, CRM Proc. Lecture Notes, vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 1–63. MR 1224050, DOI 10.1090/crmp/001/01
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
K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396, DOI 10.1007/978-3-642-52244-4
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369, DOI 10.1007/978-1-4757-2016-7
Ken Ono, On the circular summation of the eleventh powers of Ramanujan’s theta function, J. Number Theory 76 (1999), no. 1, 62–65. MR 1688192, DOI 10.1006/jnth.1998.2354
Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988.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
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
Seung Hwan Son, Circular Summation of Theta Functions in Ramanujan’s Lost Notebook, preprint.
Neil J. A. Sloane, Neil Sloane’s On-Line Encyclopedia of Integer Sequence, http://www.att.research.com/~njas/sequences/index.html.