Evaluating Six Apéry-like Series of Weight $5$
Jorge Antonio González Layja
TL;DR
This work addresses the exact evaluation of six Apéry-like series of weight $5$ that couple harmonic numbers with central binomial denominators. It develops a generating-function framework tied to $\arcsin$-type expansions and logarithmic integrals $\ln(\sin x)$ and $\ln(\cos x)$, converting the series into tractable integrals. A comprehensive set of lemmas provides arcsin-based identities, logarithmic-integral evaluations, and zeta/polylog relations that link the series to constants such as $\zeta(5)$, $\zeta(2)\zeta(3)$, and $\operatorname{Li}_5\left(\frac{1}{2}\right)$, along with powers of $\ln(2)$. The main contributions are the explicit closed forms for all six sums, enriching the landscape of Apéry-like series and uncovering deeper connections between harmonic sums, polylogarithms, and logarithmic integrals.
Abstract
The main objective of this paper is to evaluate six new Apéry-like series of weight $5$ in closed form. These series involve harmonic numbers and exhibit the characteristic reciprocal central binomial coefficient structure. Generating functions for the inverse sine and identities related to harmonic numbers are used to link each series to a variety of integrals containing $\ln \left(\sin \left(x\right)\right)$ and $\ln \left(\cos \left(x\right)\right)$, which are evaluated using a range of methods and identities.