On Central Binomial Series Related to Zeta(4)

Abstract

In this paper, we prove two related central binomial series identities: $B(4)=\sum_{n \geq 0} \frac{\binom{2n}n}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $C(4)=\sum_{n \in \mathbb{N}} \frac{1}{n^4 \binom{2n}n}=\frac{17 \pi^4}{3240}.$ 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 $C(4).$