A further q-analogue of Gosper's strange series

TL;DR

This paper introduces a new $q$-analogue of Gosper's strange series, providing an alternative proof and extending the series identities using $q$-series techniques and hypergeometric transformations.

Contribution

It presents a novel $q$-analogue of Gosper's ${}_{2}F_{1}$-series, different from Yamaguchi's, and offers simplified proofs and additional series variants using a $q$-analogue of an Abel-type summation.

Findings

Introduces a new $q$-analogue of Gosper's series.

Provides an alternative, simplified proof of Yamaguchi's $q$-analogue.

Derives multiple series variants related to hypergeometric identities.

Abstract

Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this $q$-analogue relies on three-term relations for ${}_{2}\phi_{1}$-series along with Heine's transformation of ${}_{2}\phi_{1}$-series. In this note, we introduce and prove, using a $q$-analogue of a series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of Yamaguchi's $q$-analogue, together with a ${}_{3}\phi_{2}$-series variant of Heine's $q$-analogue of Gauss's hypergeometric formula, a ${}_{6}\phi_{5}$-series variant and two ${}_{4}\phi_{3}$-series variants of the $q$-analogue of Kummer's identity due to Bailey and Daum, along with a $q$-analogue of a result obtained by Cantarini [Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}{\pi}$.