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Automatic Proof of Theta-Function Identities

Part of the Texts & Monographs in Symbolic Computation book series (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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Correspondence to Frank Garvan .

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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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