New series identities for $\frac{1}{\Pi}$
Commun. Korean Math. Soc. 2017 Vol. 32, No. 4, 865-874
Published online October 31, 2017
Mohammed M. Awad, Asmaa O. Mohammed, Medhat A. Rakha, Arjun K. Rathie
Suez Canal University, Suez Canal University, Suez Canal University, Riverside Transit Campus
Abstract : In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems have been found interesting applications in obtaining various series identities for $\Pi$, $\Pi^{2}$ and $\frac{1}{\Pi}$. The aim of this research paper is to provide twelve general formulas for $\frac{1}{\Pi}$. On specializing the parameters, a large number of very interesting series identities for $\frac{1}{\Pi}$ not previously appeared in the literature have been obtained. Also, several other results for multiples of $\Pi$, $\Pi^{2}$, $\frac{1}{\Pi^{2}}$, $\frac{1}{\Pi^{3}}$ and $\frac{1}{\sqrt{\Pi}}$ have been obtained. The results are established with the help of the extensions of classical Gauss's summation theorem available in the literature.
Keywords : hypergeometric summation theorems, Watson's theorem, Whipple's theorem, Ramanujan series for $\frac{1}{\pi}$
MSC numbers : 33C05, 33C20, 33C70
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