Summation formulas of $q$-hyperharmonic numbers

TL;DR

This paper derives new weighted summation formulas for $q$-hyperharmonic numbers, including special cases involving hyperharmonic numbers with polynomial weights, expanding the mathematical understanding of these sequences.

Contribution

It introduces novel summation formulas for $q$-hyperharmonic numbers and their special cases, enhancing the theoretical framework of hyperharmonic number identities.

Findings

Derived weighted summation formulas for $q$-hyperharmonic numbers

Obtained formulas for hyperharmonic numbers with polynomial weights

Expanded the theoretical understanding of hyperharmonic number identities

Abstract

In this paper, several weighted summation formulas of $q$-hyperharmonic numbers are derived. As special cases, several formulas of hyperharmonic numbers of type $\sum_{\ell=1}^{n} {\ell}^{p} H_{\ell}^{(r)}$ and $\sum_{\ell=0}^{n} {\ell}^{p} H_{n-\ell}^{(r)}$ are obtained.