# Some evaluation of parametric Euler sums

## TL;DR

This paper explores the analytic representations of parametric Euler sums involving harmonic numbers, providing explicit formulas and new insights using complex analysis techniques.

## Contribution

It introduces explicit formulas for parametric quadratic and cubic Euler sums in terms of zeta values and rational series, advancing the understanding of these sums.

## Key findings

- Explicit formulas for quadratic and cubic sums derived
- New consequences and examples provided
- Analytic representations using contour integrals and residues

## Abstract

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers through zeta values and rational function series, either linearly or nonlinearly. Furthermore, we give explicit formulae for several parametric quadratic and cubic sums in terms of zeta values and rational series. Moreover, some interesting new consequences and illustrative examples are considered.