Evaluations of sums involving odd harmonic numbers and binomial coefficients
For \(r,n\in \mathbb{N}\) let \(h_n^{(r)}\) denote the \(n\)th partial sum of the zeta-function \((1-2^{-r})\zeta(r)\) with \(h_n^{(1)}\) denoting the odd part of the harmonic number.\par Let \(\boldsymbol{k}=(k_1,\cdots,k_p)\in \mathbb{N}^p\) and \(\boldsymbol{\pi}=(\pi_1,\cdots,\pi_m)\in \mathbb{N}^m\), where the components are arranged in non-decreasing order and \(p, m \in \mathbb{N}\).\par Definition 1.2. The Euler \(T\)-type sum \(ET_{\boldsymbol{\pi}}^{(q)}(\boldsymbol{k})\) is defined by the convergent series \[ET_{\boldsymbol{\pi}}^{(q)}(\boldsymbol{k})=\sum_{n=1}^{\infty}\frac{n^qh_n^{\pi_1}\cdots h_n^{\pi_m}}{\prod_{i=1}^{p}\binom{n+k_1}{k_i}},\] where \(0 \le q \le k_1 + \cdots + k_p - p - 1\) and \(k_1 +\cdots+ k_p \ge 2\). The counterpart \(ET_{\boldsymbol{\pi},q}(\boldsymbol{k})\) is defined by putting the factor \(n^q\) in the numerator of the above expression.\par The main result is that the Euler \(T\)-type sums \(ET_{\boldsymbol{\pi},q}(\boldsymbol{k})\) and \(ET_{\boldsymbol{\pi}}^{(q)}(\boldsymbol{k})\) can be expressed in terms of a combination of products of \(\log 2\), zeta values, double \(T\)-values, (odd) harmonic numbers and double \(T\)-sums.
scientific article; zbMATH DE number 7857929
scientific article