$q$-Rational Reduction and $q$-Analogues of Series for $\pi$

TL;DR

This paper develops a $q$-analogue of polynomial reduction for hypergeometric terms, enabling the automatic proof of $q$-series identities related to $ppa$, and introduces new $q$-series analogues of classical $ppa$ series.

Contribution

It generalizes $q$-polynomial reduction to the rational case using $q$-Gosper representation, facilitating the discovery of $q$-series identities for $ppa$.

Findings

Derived new $q$-analogues of series for $ppa$

Extended polynomial reduction to rational functions in the $q$-hypergeometric context

Provided tools for automatic proof and discovery of $q$-identities

Abstract

In this paper, we present a $q$-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the $q$-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given $q$-hypergeometric term. The structure theorem enables us to generalize the $q$-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of $q$-identities. As applications, several $q$-analogues of series for $\pi$ are presented.