Summation formulas of hyperharmonic numbers with their generalizations II

TL;DR

This paper derives new summation formulas involving hyperharmonic numbers and their generalizations, extending classical identities and providing a broader understanding of these special number sequences.

Contribution

It introduces several new formulas for sums of hyperharmonic numbers and their generalizations, expanding upon previous harmonic number identities.

Findings

Formulas for sums of hyperharmonic numbers with powers and products

Extensions to generalized hyperharmonic numbers

New identities broadening the scope of harmonic number theory

Abstract

In 1990, Spie\ss \, gave some identities of harmonic numbers including the types of $\sum_{\ell=1}^n\ell^k H_\ell$, $\sum_{\ell=1}^n\ell^k H_{n-\ell}$ and $\sum_{\ell=1}^n\ell^k H_\ell H_{n-\ell}$. In this paper, we derive several formulas of hyperharmonic numbers including $\sum_{\ell=0}^{n} {\ell}^{p} h_{\ell}^{(r)} h_{n-\ell}^{(s)}$ and $\sum_{\ell=0}^n \ell^{p}(h_{\ell}^{(r)})^{2}$. Some more formulas of generalized hyperharmonic numbers are also shown.