The sixth, eighth, ninth, and tenth powers of Ramanujan’s theta function
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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 $F_{10}(q)$.References
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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
- Received by editor(s): July 10, 1998
- Published electronically: October 6, 1999
- Communicated by: David E. Rohrlich
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1333-1338
- MSC (1991): Primary 11B65, 33D10
- DOI: https://doi.org/10.1090/S0002-9939-99-05181-3
- MathSciNet review: 1646322