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Automatic Proof of Theta-Function Identities

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan’s 40 identities for his so called Rogers–Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for generalizations of Ramanujan’s G(q) and H(q) and Robin’s extensions. These generalizations are associated with certain real Dirichlet characters. We find a total of over 150 identities.

A preliminary version of this paper was presented by J. Frye on January 10, 2013 at JMM2013, San Diego. F. Garvan was supported in part by a grant from the Simon’s Foundation (#318714).

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Author information

Authors and Affiliations

  1. Bunker Hill Community College, Boston, MA, 02129, USA

    Jie Frye

  2. Department of Mathematics, University of Florida, Gainesville, FL, 32601, USA

    Frank Garvan

Authors
  1. Jie Frye
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  2. Frank Garvan
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Correspondence to Frank Garvan .

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Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

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Frye, J., Garvan, F. (2019). Automatic Proof of Theta-Function Identities. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_10

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  • DOI: https://doi.org/10.1007/978-3-030-04480-0_10

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  • Online ISBN: 978-3-030-04480-0

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