A q-product tutorial for a q-series MAPLE package

TL;DR

The paper presents a Maple-based toolkit for $q$-series that automates conversion to $q$-products, factorisation into finite $q$-products, and discovery of algebraic relations among $q$-series. It implements Andrews' algorithm (prodmake) for series-to-product conversion and includes modules for eta- and Jacobi-product manipulations, sifting, and classical product identities (Triple, Quintuple, Winquist). Key contributions are the practical implementations of prodmake, qfactor, etamake, jacprodmake, jacprodmake, and relation-finding tools (findhom, findhomcombo, findnonhom, findnonhomcombo, findpoly), with extensive demonstrations and exercises. This enables systematic exploration of modular forms, partitions, and theta/eta-product factorizations, facilitating both theoretical insight and computational experimentation. The package thus provides a versatile, extensible platform for symbolic computation in $q$-series and related areas of combinatorics and number theory.

Abstract

This is a tutorial for using a new q-series Maple package. The package includes facilities for conversion between q-series and q-products and finding algebraic relations between q-series. Andrews found an algorithm for converting a q-series into a product. We provide an implementation. As an application we are able to effectively find finite q-product factorisations when they exist thus answering a question of Andrews. We provide other applications involving factorisations into theta functions and eta products.