A $q$-analogue of Gosper's strange evaluation of the hypergeometric series

TL;DR

This paper introduces a $q$-analogue of a hypergeometric series identity originally conjectured by Gosper, providing a generalized form using basic hypergeometric series relations.

Contribution

The paper presents a new $q$-analogue of Gosper's hypergeometric identity and its generalization, expanding the understanding of basic hypergeometric series.

Findings

Derived a $q$-analogue of Gosper's identity

Generalized the $q$-analogue using three-term relations

Extended the identity to a broader class of series

Abstract

In 1977, Gosper conjectured many strange evaluations of hypergeometric series. One of them is a ${}_{2}F_{1}$-series identity with two free parameters, which was proved by Ebisu (2013), Chu (2017), and Campbell (2023) in different ways. In this paper, we present a $q$-analogue of the ${}_{2}F_{1}$-series identity, along with its generalization, by using three-term relations for the ${}_{2}\phi_{1}$ basic hypergeometric series.