Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments

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.

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