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

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.