Another look at Zagier's formula for multiple zeta values involving Hoffman elements

TL;DR

This paper provides an elementary proof of Zagier's formula involving Hoffman elements for multiple zeta values, aiding in Hoffman's conjecture and deriving new formulas for Hurwitz zeta values.

Contribution

It offers a direct proof of Zagier's formula using rational zeta series and introduces a new Zagier-type formula for multiple special Hurwitz zeta values.

Findings

Elementary proof of Zagier's formula involving Hoffman elements

Verification of Hoffman's conjecture for multiple zeta values

New Zagier-type formula for multiple special Hurwitz zeta values

Abstract

In this paper, we give an elementary account into Zagier's formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient $\zeta(2n)$. This formula plays an important role in proving Hoffman's conjecture which asserts that every multiple zeta value of weight $k$ can be expressed as a $\mathbb{Q}$-linear combinations of multiple zeta values of the same weight involving $2$'s and $3$'s. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values.