Q-Rational reduction and q-analogues of series for π

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 π are presented.