On Some Classes of Series Representations for 1/π1/π1/\pi and π2π2\pi^2

Hakan KÜÇÜK; Sezer SORGUN

doi:https://doi.org/10.36753/mathenot.795582

Makale detayları

On Some Classes of Series Representations for 1/π1/π1/\pi and π2π2\pi^2

Yazarlar:

Hakan KÜÇÜK | Sezer SORGUN

Yayıncı:

Murat TOSUN

Ana Başlıklar:

Matematik

Dergi:

Mathematical Sciences and Applications E-Notes 2021 Vol.9 Issue 4, pp.176-184

Belge Tipi:

Makale

Yayın Tarihi:

12/2021

Belge Dili:

İngilizce

Document ID (Org No.):

2000541629

Sıklık:

Üç Aylık

eISSN:

2147-6268

DOI:

https://doi.org/10.36753/mathenot.795582

Özet:

 

We propose some classes of series representations for 1/π1/π1/\pi and π2π2\pi^2 by using a new WZ-pair. As examples, among many others,  we prove that
32∞∑n=1n16n(n+1)(2n−1)(2nn)2=1π,32∑n=1∞n16n(n+1)(2n−1)(2nn)2=1π,\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*}
1−14∞∑n=03n+2(n+1)2(2nn)2116n=1π1−14∑n=0∞3n+2(n+1)2(2nn)2116n=1π\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∞∑n=01(n+1)(2n+1)4n(2nn)=π2.4∑n=0∞1(n+1)(2n+1)4n(2nn)=π2.
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.

Makale detayları

On Some Classes of Series Representations for 1/π1/π1/\pi and π2π2\pi^2

Yazarlar: Hakan KÜÇÜK | Sezer SORGUN

Yayıncı: Murat TOSUN

Ana Başlıklar: Matematik

Dergi: Mathematical Sciences and Applications E-Notes 2021 Vol.9 Issue 4, pp.176-184

Belge Tipi: Makale

Yayın Tarihi: 12/2021

Belge Dili: İngilizce

Document ID (Org No.): 2000541629

Sıklık: Üç Aylık

eISSN: 2147-6268

DOI: https://doi.org/10.36753/mathenot.795582

Sayı

Yazarlar:

Hakan KÜÇÜK | Sezer SORGUN

Yayıncı:

Murat TOSUN

Ana Başlıklar:

Matematik

Dergi:

Mathematical Sciences and Applications E-Notes 2021 Vol.9 Issue 4, pp.176-184

Belge Tipi:

Makale

Yayın Tarihi:

12/2021

Belge Dili:

İngilizce

Document ID (Org No.):

2000541629

Sıklık:

Üç Aylık

eISSN:

2147-6268

DOI:

https://doi.org/10.36753/mathenot.795582