On generalized harmonic numbers, Tornheim double series and linear Euler sums

TL;DR

This paper explores the relationships between generalized harmonic numbers, Euler sums, and Tornheim series, providing new decompositions, evaluations, and reductions to zeta values to deepen understanding of these mathematical objects.

Contribution

It establishes clearer links between harmonic numbers, Euler sums, and Tornheim series, offering new evaluations and reductions to zeta values.

Findings

Every linear Euler sum can be expressed as a combination of Tornheim series.

New closed-form formulas for Euler sums are derived.

Certain Euler sum combinations are reducible to Riemann zeta values.

Abstract

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.