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

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.