On Some Classes of Series Representations for $1/\pi$ and $\pi^2$
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^n(n+1)(2n-1)}\binom{2n}{n}^2=\frac{1}{\pi},
\end{equation*}
\begin{equation*}
1-\frac{1}{4}\sum_{n=0}^{\infty}\frac{3n+2}{(n+1)^2}\binom{2n}{n}^2 \frac{1}{16^n}=\frac{1}{\pi}
\end{equation*}
and
$$
4\sum_{n=0}^{\infty}\frac{1}{(n+1)(2n+1)}\frac{4^n}{ \binom{2 n}{n}}=\pi^2.
$$
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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