On Some Classes of Series Representations for $1/pi$ and $pi^2$

Authors : Hakan KÜÇÜK, Sezer SORGUN

Pages : 176-184

Doi:10.36753/mathenot.795582

View : 40 | Download : 11

Publication Date : 2021-12-31

Article Type : Research Paper

Abstract :We propose some classes of series representations for $1/\pi$ and $\pi^2$ by using a new WZ-pair. As examples, among many others, we prove that \begin{equation*} \frac{3}{2}\sum_{n=1}^{\infty}\frac{n}{16^ninsert ignore into journalissuearticles values(n+1);insert ignore into journalissuearticles values(2n-1);}\binom{2n}{n}^2=\frac{1}{\pi}, \end{equation*} \begin{equation*} 1-\frac{1}{4}\sum_{n=0}^{\infty}\frac{3n+2}{insert ignore into journalissuearticles values(n+1);^2}\binom{2n}{n}^2 \frac{1}{16^n}=\frac{1}{\pi} \end{equation*} and $$ 4\sum_{n=0}^{\infty}\frac{1}{insert ignore into journalissuearticles values(n+1);insert ignore into journalissuearticles values(2n+1);}\frac{4^n}{ \binom{2 n}{n}}=\pi^2. $$ Furthermore, our results lead to new combinatorial identities and binomial sums involving harmonic numbers.
Keywords : Ramanujan type series, WZ pair, Combinatorial identities, Binomial sums, Ekhad package

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