$q$-Analogues of $\pi$-Related Formulae from Jackson's $_8\phi_7$-Series via Inversion Approach

Abstract

By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $\pi$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/\pi^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfloor{\frac{n}3}\rfloor+\lfloor{\frac{n+1}3}\rfloor+\lfloor{\frac{n+2}3}\rfloor$ are examined via the ``reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts of convergence rate ``$\frac{-1}{27}$" when $q\to1$.