Multiple zeta values and multiple Ap\'ery-like sums

TL;DR

This paper introduces Apéry-like sums, shows they can express all multiple zeta values, and provides new formulas and identities, advancing understanding of their structure and relations.

Contribution

It formally defines Apéry-like sums, demonstrates their ability to represent all multiple zeta values, and offers new integral formulas and identities.

Findings

All multiple zeta values can be expressed as integer combinations of Apéry-like sums.

New integral formulas for multiple zeta values and Apéry-like sums are derived.

Short proofs of known formulas, like Zagier's, are provided using these new representations.

Abstract

In this paper, we formally introduce the notion of Ap{\'e}ry-like sums and we show that every multiple zeta values can be expressed as a $\bf Z$-linear combination of them. We even describe a canonical way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Ap\'ery-like sums. They enable us to give a short direct proof of Zagier's formulas for $\zeta(2,\ldots,2,3,2,\ldots,2)$ as well as of similar ones in the context of Ap{\'e}ry-like sums. The relations between Ap\'ery-like sums themselves still remain rather mysterious, but we get significant results and state some conjectures about their pattern.