During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{ k=1}^{ \infty} \frac{ (-1)^k}{ k^2{ 2k \choose k}{ 3k \choose k}} \left( \frac{ 7 k-2}{ 2 k-1} H_{ k-1}^{ (2)}-\frac{ 3}{ 4 k^2} \right) = \frac{ \pi^4}{ 720}. \] and \[ \sum_{ k=1}^\infty \frac{ 1}{ k^2 { 2k \choose k}^2} \left( \frac{ 30k-11}{ k(2k-1)} (H_{ 2k-1}^{ (3)} + 2 H_{ k-1}^{ (3)}) + \frac{ 27}{ 8k^4} \right) = 4 \zeta(3)^2, \] where $H_n^{ (m)}$ denotes $\sum_{ 0<j \le n}j^{ -m}$.
Categories: math.CO