Euler sums of generalized harmonic numbers and connected extensions

TL;DR

This paper evaluates Euler sums involving generalized hyperharmonic numbers and expresses them in terms of classical Euler sums, also deriving results for series with harmonic numbers and reciprocal binomial coefficients.

Contribution

It introduces new evaluations of Euler sums of generalized hyperharmonic numbers and related series in terms of Riemann zeta values.

Findings

Euler sums of hyperharmonic numbers expressed via classical Euler sums

Evaluation of series with harmonic numbers and reciprocal binomial coefficients

Connections established between hyperharmonic sums and Riemann zeta values

Abstract

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}% \] in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of Riemann zeta values.