Basic hypergeometric identities derived from three-term relations

Abstract

In 2015, Ebisu presented a new method for finding hypergeometric identities based on three-term relations for the ${}_{2} F_{1}$ hypergeometric series. By using this method, he derived almost all of the previously known hypergeometric identities, as well as many new ones. In this paper, we derive several basic hypergeometric identities, including both well-known and not widely known ones, by applying a $q$-analogue of Ebisu's method to three-term relations for the ${}_{2} \phi_{1}$ basic hypergeometric series.