Automatic Proof of Theta-Function Identities

TL;DR

This paper introduces two MAPLE packages, thetaids and ramarobinsids, for proving and discovering theta-function and eta-product identities, including Ramanujan's identities and their generalizations, using modular function techniques.

Contribution

It presents new computational tools for automatically proving and finding theta-function identities, extending Ramanujan's work to higher-level eta-products and Dirichlet character generalizations.

Findings

Over 150 identities found and proved

Automated verification of Ramanujan's identities

Tools applicable to generalized eta-products and modular functions

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.