A further q-analogue of a formula due to Guillera

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.