Parametric Euler sums of harmonic numbers (Q6614561)

Let \(n\) and \(p\) be positive integers. The generalized harmonic number \(H_n^{(p)}\) is defined by \[ H_n^{(p)}=\sum_{k=1}^n \frac{1}{k^p}.\] In this paper under review , the authors use the approach of contour integral representation to give explicit evaluations for some of the following (alternating) parametric Euler sums involving generalized harmonic numbers \[S_{p, q}^\sigma (a_1,\ldots,a_m)=\sum_{n=1}^\infty \frac{H_n^{(p_1)}\cdots H_n^{(p_r)}\sigma^n }{(n+a_1)^{q_1}\cdots (n+a_m)^{q_m}}, \] where \(a_1,\ldots, a_m\) are non negative integers, \(p_1,\ldots, p_r \in \mathbb{N}^*\), \(q_1,\ldots, q_m \in\mathbb{N}\) with \(q_1+\cdots+q_m \geq 2\) and \(\sigma=\pm1\). For this purpose, the authors use a similar method of countour integration due to \textit{P. Flajolet} and \textit{B. Salvy} [Exp. Math. 7, No. 1, 15--35 (1998; Zbl 0920.11061)] in a more recent paper of \textit{C. Xu} [J. Math. Anal. Appl. 451, No. 2, 954--975 (2017; Zbl 1405.11110)].\N\NFinally, the authors also give an explicit evaluation of alternating double zeta values \(\zeta (\overline{2j};2m+1)\) in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.