Series with summands involving harmonic numbers

Abstract

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum _{0<k\le n}\frac{1}{k^m}(n=0,1,2,\dots ).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum _{k=0}^{\infty }(6k+1)\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}{H}_k^{(3)}\right)=\frac{25\zeta (3)}{8\pi }-G,$$ 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.

41 pages, final version

Cited by in corpus (1)

• Sun's Series via Cyclotomic Multiple Zeta Values