$q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach

Abstract

UDC 517.588, 517.52

By using the multiplicative form of the extended Carlitz inverse-series relations, we establish two general ``dual'' theorems on Jackson's summation formula for the well-poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n = \Big\lfloor\dfrac{n}2 \Big\rfloor + \Big\lfloor\dfrac{n + 1}2 \Big\rfloor $ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in the classical $\pi$-related Ramanujan-like series with convergence rate ``"$dfrac1{16}$", including the series for $1/\pi^2$ discovered by Guillera (2003). The triplicate dual formulas under the partition pattern $n = \Big\lfloor{\dfrac{n}3}\Big\rfloor + \Big\lfloor{\dfrac{n + 1}3} \Big\rfloor + \Big\lfloor{\dfrac{n + 2}3} \Big\rfloor $ are examined via the "reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts with the convergence rate "$\dfrac{-1}{27}$" as $q\to1.$