$\pi$-Formulae from dual series of the Dougall theorem

W.  Chu

doi:10.37863/umzh.v74i12.6587

W. Chu School Math. and Statistics, Zhoukou Normal Univ., Henan, China and Univ. Salento, Italy

DOI: https://doi.org/10.37863/umzh.v74i12.6587

Keywords: Classical hypergeometric series; The Dougall summation theorem; Gould-Hsu inverse series relations; Ramanujan's series for $1/\pi$; Guillera's series for $1/\pi^2$.

Abstract

UDC 517.5

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

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