A further $q$-analogue of a formula due to Guillera

TL;DR

This paper introduces a new $q$-analogue of Guillera's $rac{ ext{pi}^2}{6}$ series, featuring an additional free parameter, derived using the $q$-Zeilberger algorithm, expanding the landscape of $q$-series identities.

Contribution

The paper presents a novel $q$-analogue of Guillera's formula that includes an extra free parameter, differing from previous known $q$-analogues, and employs the $q$-Zeilberger algorithm for its derivation.

Findings

Introduces a new $q$-analogue with an extra free parameter.

Proves the new $q$-analogue using the $q$-Zeilberger algorithm.

Demonstrates the inequivalence of the new $q$-analogue to previous ones.

Abstract

Hou, Krattenthaler, and Sun have introduced two $q$-analogues of a remarkable series for $\pi^2$ due to Guillera, and these $q$-identities were, respectively, proved with the use of a $q$-analogue of a Wilf-Zeilberger pair provided by Guillera and with the use of ${}_{3}\phi_{2}$-transforms. We prove a $q$-analogue of Guillera's formula for $\pi^2$ that is inequivalent to previously known $q$-analogues of the same formula due to Guillera, including the Hou-Krattenthaler-Sun $q$-identities and a subsequent $q$-identity due to Wei. In contrast to previously known $q$-analogues of Guillera's formula, our new $q$-analogue involves another free parameter apart from the $q$-parameter. Our derivation of this new result relies on the $q$-analogue of Zeilberger's algorithm.