On some conjectural series containing harmonic numbers of 3-order

TL;DR

This paper proves four series involving third-order harmonic numbers using advanced hypergeometric series techniques, confirming three conjectures by Z.-W. Sun.

Contribution

It introduces novel proofs for four series with third-order harmonic numbers, including three conjectures by Sun, using derivative, integral, and transformation methods.

Findings

Proved four series involving third-order harmonic numbers.

Confirmed three conjectures proposed by Z.-W. Sun.

Applied hypergeometric series techniques to harmonic number identities.

Abstract

Harmonic numbers are important in a lot of branches of number theory. By means of the derivative operator, the integral operator, and several summation and transformation formulas for hypergeometric series, we prove four series containing harmonic numbers of 3-order. Three of them are conjectures which were recently proposed by Z.-W. Sun.