Parametric Euler Sums of Harmonic Numbers

TL;DR

This paper introduces a parametric approach to generalized Euler sums involving harmonic numbers, providing explicit formulas and evaluations for various zeta functions and alternating double zeta values using contour integration.

Contribution

It develops a new parametric framework for Euler sums and derives explicit formulas for zeta functions and alternating double zeta values, advancing the analytical understanding of these sums.

Findings

Explicit formulas for (Hurwitz) zeta functions

Evaluation of linear and quadratic parametric Euler sums

Explicit evaluation of alternating double zeta values

Abstract

We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values $\ze(\overline{2j},2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.