Welcome to mathimatikoi.org forum; Enjoy your visit here.

A closed form of a hypergeometric series

Calculus (Integrals, Series)
Post Reply
User avatar
Tolaso J Kos
Administration team
Administration team
Articles: 2
Posts: 855
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa
Contact:

A closed form of a hypergeometric series

#1

Post by Tolaso J Kos » Wed Aug 16, 2017 12:09 pm

The following result is new and is going to be published on Arxiv.org in the upcoming days with many more interesting results by Jacopo D' Aurizio who has actually proved it. Nevertheless , I am posting it here since it is interesting , challenging as well as approachable using only elementary tools.

Prove that

\begin{align*}
{}_4 F_3 \left ( 1, 1, 1 , \frac{3}{2} ; \frac{5}{2} , \frac{5}{2} , \frac{5}{2} ; 1 \right ) &= 27 \sum_{n=0}^{\infty} \frac{16^n}{\left ( 2n+3 \right )^3 \left ( 2n+1 \right )^2 \binom{2n}{n}^2} \\
&= \frac{27}{2} \bigg( 7 \zeta(3) + \left ( 3 - 2 \mathcal{G} \right ) \pi - 12 \bigg)
\end{align*}

where $\mathcal{G}$ denotes the Catalan's constant.
Imagination is much more important than knowledge.
Post Reply