Combinatorial identities involving harmonic numbers

TL;DR

This paper proves a new combinatorial identity and uses it to derive numerous finite harmonic sum identities, many of which are novel contributions to the mathematical literature.

Contribution

The work introduces a new combinatorial identity and applies it to establish many new finite harmonic sum identities, expanding the understanding of harmonic number relations.

Findings

Derived new harmonic sum identities involving binomial coefficients and harmonic numbers.

Most results are original, with some recapturing known identities.

Provides a framework for generating further harmonic sum identities.

Abstract

In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*} \sum_{k=1}^{n}\frac{(-1)^{k-1}}{k}\binom{n}{k}H_{n-k}=H_n^2+\sum_{k=1}^{n}\frac{(-1)^{k}}{k^2\binom{n}{k}}, \end{equation*} and \begin{equation*} \sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}\binom{n}{k}H_{n-k}=\frac{H_n[H_n^2+H_n^{(2)}]}{2}-\sum_{k=0}^{n-1}\frac{(-1)^k[H_n-H_k]}{(k+1)(n-k)\binom{n}{k}}. \end{equation*} Almost all of our results are new, while a few of them recapture know results.