Two variants of Euler sums (Q2172596)

\textit{P. Flajolet} and \textit{B. Salvy} [Exp. Math. 7, No. 1, 15--35 (1998; Zbl 0920.11061)] defined the classical Euler sums: \[ S_{p_1p_2\cdots p_k,q}:=\sum_{n=1}^{\infty}\frac{H_n^{(p_1)}H_n^{(p-2)}\cdots H_n^{(p_k)}}{n^q}, \] where \(p_1,p_2,\dots, p_k\in \mathbb{N}\), with \(p_1\leq p_2\leq \dots \leq p_k\), and \(2\leq q\in \mathbb{N}\). Here \[ H_n^{(p)}:=\sum_{k=1}^n\frac{1}{k^p}. \] The classical Euler sums have been studied extensively from many aspects: combinatorial, analytic and algebraic. The classical Euler sums are also closely related to the theory of multiple zeta values. In this paper, the authors defined the following two variants of Euler sums: \[ T_{p_1p_2\cdots p_k,q}:=\sum_{n=1}^{\infty}\frac{h_{n-1}^{(p_1)}h_{n-1}^{(p_2)}\cdots h_{n-1}^{(p_k)} }{(n-1/2)^q }, \text{ for }q\geq 2, \] \[ T_{p_1p_2\cdots p_k,\widetilde{q}}:=\sum_{n=1}^{\infty}(-1)^n\frac{h_{n-1}^{(p_1)}h_{n-1}^{(p_2)}\cdots h_{n-1}^{(p_k)} }{(n-1/2)^q }, \text{ for }q\geq 2, \] where \[ h_n^{(p)}:=\sum_{k=1}^n\frac{1}{(k-1/2)^p}. \] They established many formulas for evaluations of the above two variants of Euler sums in many special cases. They also show the relations among the two variants of Euler sums , Hoffman \(t\)-values and Kaneko-Tsumura \(T\)-values.