Explicit Evaluations for Several Variants of Euler Sums

TL;DR

This paper investigates various Euler sum variants using complex analysis techniques, revealing their properties and relationships with a new level 2 multiple zeta value variant, advancing understanding of special number series.

Contribution

It introduces new variants of Euler sums and a level 2 multiple zeta value, providing identities and properties that extend classical results.

Findings

Variants exhibit closed forms and reduction properties.

Identities relating Euler sum variants and level 2 multiple zeta values.

New relationships between different special number series.

Abstract

We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.