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http://dx.doi.org/10.4134/CKMS.c160194

NEW SERIES IDENTITIES FOR ${\frac{1}{\Pi}}$  

Awad, Mohammed M. (Department of Mathematics Faculty of Science Suez Canal University)
Mohammed, Asmaa O. (Department of Mathematics Faculty of Science Suez Canal University)
Rakha, Medhat A. (Department of Mathematics Faculty of Science Suez Canal University)
Rathie, Arjun K. (Department of Mathematics School of Mathematical and Physical Sciences Central University of Kerala Riverside Transit Campus)
Publication Information
Communications of the Korean Mathematical Society / v.32, no.4, 2017 , pp. 865-874 More about this Journal
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 ${\frac{1}{\Pi}}$;
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