Taylor coefficients and series involving harmonic numbers

TL;DR

The paper addresses Sun's conjectures on infinite series with harmonic-number summands by combining two complementary strategies: (i) extracting Taylor coefficients of parametric hypergeometric identities via Wilf–Zeilberger (WZ) pairs to produce harmonic-number relations, and (ii) applying hypergeometric transformations and differential operations to generate a wide range of identities. It reports 58 identities, including eight conjectures, and connects results to constants such as $\pi$, $\zeta(n)$, Catalan's constant $G$, and $\beta(4)$, among others. The work expands the toolkit for proving Sun's conjectures and demonstrates how WZ methods and hypergeometric transformations can be fused to yield high-signal harmonic-number identities with explicit closed forms. Overall, the findings advance the understanding of series involving generalized harmonic numbers and their connections to fundamental constants in number theory and analysis.

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 \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{π^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 ζ(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0<j \le n}j^{-m}$.

Theorems & Definitions (23)

• Theorem 1.1
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• Theorem 1.10