On Evaluations of Euler-type Sums of Hyperharmonic Numbers

TL;DR

This paper provides explicit formulas for Euler-type sums involving hyperharmonic numbers, extending known results to arbitrary integer orders and expressing them through zeta values, harmonic numbers, and Euler sums.

Contribution

It introduces new explicit evaluations of Euler sums of hyperharmonic numbers for any integer r, expanding the scope of closed-form formulas.

Findings

Explicit evaluations of linear and non-linear Euler sums of hyperharmonic numbers.

Extension of closed-form formulas to arbitrary integer r.

Formulas expressed in terms of zeta values, harmonic numbers, and Euler sums.

Abstract

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer $r$. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums.