On Some Series Involving Harmonic and Skew-Harmonic Numbers
Vincent Nguyen
TL;DR
This paper provides closed-form evaluations for two classes of harmonic-related series: tails involving $H_n$ and skew-harmonic numbers, and Hardy-type sums where a logarithmic correction is subtracted from $H_n$. It employs integral representations, Abel summation, and deep connections between zeta values, Dirichlet eta, polylogarithms, and special functions (including the Barnes $G$-function and the Euler-Gompertz constant) to derive exact expressions for tail sums $S_1$ and $S_2$, and to present a general Hardy-series framework with rich corollaries. Key results include $S_1 = \log(2) - \frac{7}{8}\zeta(3) - 1$, a closed form for $S_2$ in terms of multiple zeta values and polylog constants, and a general Hardy-type identity with corollaries such as $x=1$ and $k=1$, plus a broader theorem involving the Barnes $G$-function. This work strengthens the links between Euler sums, polylogarithms, and special constants, offering precise tools for analytic number theory and symbolic computation.
Abstract
In this paper, we evaluate in closed form several different series involving the harmonic numbers and skew-harmonic numbers. We consider two classes of series involving these sequences. One class of series involves the product of the $n$th harmonic or skew-harmonic number and a tail. We provide the solution to two open problems concerning these harmonic series with tails from Ovidiu Furdui's book Sharpening Mathematical Analysis Skills. The other class of series is the Hardy series, which involves a logarithm and the Euler-Mascheroni constant being subtracted from the $n$th harmonic number.