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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 sixth, eighth, ninth, and tenth powers
of Ramanujan's theta function


Author: Scott Ahlgren
Journal: Proc. Amer. Math. Soc. 128 (2000), 1333-1338
MSC (1991): Primary 11B65, 33D10
Published electronically: October 6, 1999
MathSciNet review: 1646322
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Abstract: In his Lost Notebook, Ramanujan claimed that the ``circular'' summation of the $n$-th powers of the symmetric theta function $f(a,b)$ satisfies a factorization of the form $f(a,b)F_{n}(ab)$. Moreover, Ramanujan recorded identities expressing $F_{2}(q)$, $F_{3}(q)$, $F_{4}(q)$, $F_{5}(q)$, and $F_{7}(q)$ in terms of his theta functions $\varphi (q)$, $\psi (q)$, and $f(-q)$. Ramanujan's claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for $F_{6}(q)$, $F_{8}(q)$, $F_{9}(q)$, and $F_{10}(q)$.


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Additional Information

Scott Ahlgren
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802-6401
Address at time of publication: Department of Mathematics, Colgate University, Hamilton, New York 13346
Email: ahlgren@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05181-3
PII: S 0002-9939(99)05181-3
Keywords: Ramanujan, theta functions
Received by editor(s): July 10, 1998
Published electronically: October 6, 1999
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society