Series with summands involving harmonic numbers

Abstract

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum_{k=0}^\infty(6k+1)\frac{\binom{2k}k^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}H_{k}^{(3)}\right) =\frac{25\zeta(3)}{8\pi}-G,$$ where $G$ denotes the Catalan constant $\sum_{k=0}^\infty(-1)^k/(2k+1)^2$. This paper contains $70$ conjectures posed by the author during 2022--2023.