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Differential equations for septic theta functions

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Abstract

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas - Pan American, 1201 West University Avenue, Edinburg, TX, 78539, USA

    Tim Huber & Danny Lara

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  1. Tim Huber
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  2. Danny Lara
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Correspondence to Tim Huber.

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The second author is partially supported by NSF-LSAMP Grant HRD-1202008.

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Huber, T., Lara, D. Differential equations for septic theta functions. Ramanujan J 38, 211–226 (2015). https://doi.org/10.1007/s11139-014-9588-1

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  • DOI: https://doi.org/10.1007/s11139-014-9588-1

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