# Alternating Double Euler Sums, Hypergeometric Identities and a Theorem of Zagier

## TL;DR

This paper explores relations between alternating double Euler sums and hypergeometric identities, providing new proofs and expressing certain sums in terms of zeta values, advancing understanding of multiple zeta values.

## Contribution

It introduces new relations between generating functions of double shuffle and stuffle relations, and offers novel proofs of hypergeometric identities and Zagier's formula.

## Key findings

- Expressed alternating double Euler sums in terms of zeta values.
- Provided a direct proof of a hypergeometric identity related to Andrews.
- Gave an alternative proof of Zagier's formula for specific multiple zeta values.

## Abstract

In this work, we derive relations between generating functions of double stuffle relations and double shuffle relations to express the alternating double Euler sums $\zeta\left(\overline{r}, s\right)$, $\zeta\left(r, \overline{s}\right)$ and $\zeta\left(\overline{r}, \overline{s}\right)$ with $r+s$ odd in terms of zeta values. We also give a direct proof of a hypergeometric identity which is a limiting case of a basic hypergeometric identity of Andrews. Finally, we gave another proof for the formula of Zagier on the multiple zeta values $\zeta(2,\ldots,2,3,2,\ldots,2)$.