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 $\frac{1}{\pi}$