A Family of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

TL;DR

This paper introduces a new family of identities for multiple harmonic sums, generalizing previous results, and applies them to relate multiple zeta star values to alternating Euler sums, revealing structural patterns based on string blocks.

Contribution

It presents a novel family of identities for multiple harmonic sums and connects multiple zeta star values with alternating Euler sums, extending existing mathematical frameworks.

Findings

New identities for multiple harmonic sums are established.

Relationships between multiple zeta star values and alternating Euler sums are derived.

The depth of Euler sums depends only on the number of 2-string blocks, not their length.

Abstract

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.