$\pi$-Formulae from Dual Series of the Dougall Theorem

Abstract

By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised ${}_7{F}_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for ${\pi }^{\pm 1}$ and ${\pi }^{\pm 2}$, including an elegant formula of ${\pi }^{-2}$ due to Guillera.