Inverse of a matrix related to double zeta values of odd weight

TL;DR

This paper proves Zagier's conjecture on the inverse matrix related to double zeta values of mixed parity, leading to new Bernoulli number identities and their generalizations involving negative binomial coefficients.

Contribution

It provides a proof of Zagier's conjecture and introduces a generalized family of Bernoulli number identities involving negative binomial coefficients.

Findings

Proof of Zagier's conjecture on inverse matrices related to double zeta values

Derivation of new Bernoulli number identities

Generalization involving binomial coefficients with negative arguments

Abstract

In this paper, we give a proof of a conjecture made by Zagier about the inverse of some matrix related to double zeta values of parity $(\mathrm{even},\mathrm{odd})$. As a result, we obtain a family of Bernoulli number identities. We further generalize this family to a more general setting involving binomial coefficients of negative arguments.