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

TL;DR

This paper uses inverse series relations to derive new Ramanujan-like infinite series for powers of pi, including a notable formula for pi^{-2} by Guillera, expanding the mathematical understanding of pi representations.

Contribution

It introduces a novel method leveraging dual relations of Dougall's theorem to generate new series expressions for powers of pi.

Findings

Derived new series for pi and its powers using dual series relations.

Included an elegant formula for pi^{-2} by Guillera.

Expanded the set of known Ramanujan-like series for pi.

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 $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for $\pi^{\pm1}$ and $\pi^{\pm2}$, including an elegant formula of $\pi^{-2}$ due to Guillera.