QEta package

Abstract

The QEta package is a collection of programs written in the FriCAS computer algebra system that allow to compute with Dedekind eta-functions and related \(q\)-series where \(q=e^{2\pi i \tau}\).

Furthermore, we provide a number of functions connected to the theory of modular functions.

Overview

The QEta package started with an implementation of the AB algorithm from the article An algorithmic approach to Ramanujan-Kolberg identities by Silviu Radu and the Samba algorithm from the article Dancing Samba with Ramanujan Partition Congruences by Ralf Hemmecke in addition it implements the algorithm from the article Construction of all Polynomial Relations among Dedekind Eta Functions of Level N by Hemmecke and Radu to compute all polynomial relations of Dedekind eta-functions of a certain level.

The underlying theory of the programs is described in the above articles which are also available as RISC reports 15-14, 16-06, 18-03.

Further material is in the git repository of the project.

This package requires a version of FriCAS that is compiled from at least SVN revision 2328, ie, where Gröbner basis computations do no longer require variable names.

The scripts usually use:

)set output linear on

which is coded in the file 1d.spad in the master-hemmecke branch of a clone of the FriCAS git repository. However, this linear output form is not absolutely necessary.