qFunctions -- A Mathematica package for $q$-series and partition theory applications

TL;DR

The qFunctions Mathematica package provides experimental and symbolic tools for $q$-series and partition theory, enabling automated manipulations, equation solving, and extensions of classical methods like weighted words and cylindric partitions.

Contribution

It introduces a comprehensive Mathematica package that automates symbolic and experimental analysis of $q$-series and partition problems, including new implementations of classical methods.

Findings

Includes guessers for $q$-shift equations and recurrences

Automates creation of recurrence systems for weighted words and cylindric partitions

Handles formal manipulations of $q$-differential, $q$-shift equations, and recurrences

Abstract

We describe the qFunctions Mathematica package for $q$-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for $q$-shift equations and recurrences for given $q$-series and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on $q$-differential, $q$-shift equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. Here, we also extend the classical method of the weighted words approach. Moreover, qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions.