Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments

TL;DR

This paper derives explicit formulas for certain infinite series involving harmonic numbers with rational arguments, expressed through well-known special functions, advancing the understanding of harmonic sum evaluations.

Contribution

It provides new explicit evaluations of Euler sums with harmonic numbers at rational points, involving Riemann and Hurwitz zeta functions, for the first time.

Findings

Explicit formulas for series with harmonic numbers at rational points.

Connections between harmonic sums and zeta functions.

Results applicable to series with odd sum of parameters.

Abstract

This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1, \end{equation*} for odd values of $p+q$. These explicit evaluations are expressed in terms of the Riemann zeta function and the Hurwitz zeta function.