Combinatorial identities involving harmonic numbers

Necdet Batir

Published 2018-06-08Version 1

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.