Basic hypergeometric identities derived from three-term relations

TL;DR

This paper extends Ebisu's method for deriving hypergeometric identities to basic hypergeometric series, successfully obtaining numerous known and new identities through a $q$-analogue approach.

Contribution

It introduces a $q$-analogue of Ebisu's method to derive basic hypergeometric identities from three-term relations, expanding the toolkit for hypergeometric series analysis.

Findings

Derived several basic hypergeometric identities, including well-known and novel ones.

Demonstrated the effectiveness of the $q$-analogue method in generating identities.

Enhanced understanding of three-term relations in basic hypergeometric series.

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.