$q$-Analogues of $\pi$-Series by Applying Carlitz Inversions to $q$-Pfaff-Saalsch{\"u}tz Theorem

TL;DR

This paper derives twenty-five new nonterminating $q$-series identities, including $q$-analogues of classical series for $\

Contribution

It introduces novel $q$-series identities by applying Carlitz inverse relations to the $q$-Pfaff-Saalschütz theorem, expanding the understanding of $q$-analogues of $\

Findings

Established 25 nonterminating $q$-series identities.

Some identities serve as $q$-analogues of Ramanujan and Guillera series for $\

Demonstrated the effectiveness of Carlitz inversions in deriving $q$-series identities.

Abstract

By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalsch{\"u}tz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $\pi$ and $1/\pi$, including some typical ones discovered by Ramanujan (1914) and Guillera.