On Central Binomial Series Related to Zeta(4)

Abstract

In this paper, we prove two related central binomial series identities: $B(4)={\sum }_{n\ge 0}\frac{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}{2^{4n}(2n+1{)}^3}=\frac{7{\pi }^3}{216}$ and $C(4)={\sum }_{n\in \mathbb{N}}\frac{1}{n^4\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}=\frac{17{\pi }^4}{3240}.$ Both series resist all the standard approaches used to evaluate other well-known series. To prove the first series identity, we will evaluate a log-sine integral that is equal to $B(4).$ Evaluating this log-sine integral will lead us to computing closed forms of polylogarithms evaluated at certain complex exponentials. To prove the second identity, we will evaluate a double integral that is equal to $C(4).$ Evaluating this double integral will lead us to computing several polylogarithmic integrals, one of which has a closed form that is a linear combination of $B(4)$ and $C(4).$ After proving these series identities, we evaluate several challenging logarithmic and polylogarithmic integrals, whose evaluations involve surprising appearances of integral representations of $B(4)$ and $C(4).$ We also provide an insight into the generalization of a modern double integral proof of Euler's celebrated identity ${\sum }_{n\in \mathbb{N}}\frac{1}{n^2}=\frac{{\pi }^2}{6},$ in which we encounter an integral representation of $C(4).$

References in corpus (1)

• Evaluation of Harmonic Sums with Integrals