Taylor coefficients and series involving harmonic numbers

TL;DR

This paper derives 58 series identities involving harmonic numbers, including some conjectured by Z.-W. Sun, connecting series with special constants like pi and zeta values.

Contribution

The authors establish numerous new series identities involving harmonic numbers, expanding the understanding of series related to special constants and confirming some of Sun's conjectures.

Findings

Derived 58 new series identities involving harmonic numbers.

Confirmed 8 conjectured series identities by Z.-W. Sun.

Connected series to constants like pi^4/720 and zeta(3)^2.

Abstract

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 $\sum_{0<j \le n}j^{-m}$.