Alternating Euler sums and special values of Witten multiple zeta function attached to so(5)

TL;DR

This paper investigates the special values of the Witten multiple zeta function associated with so(5), demonstrating they can be expressed using alternating Euler sums, with specific exceptions involving the Riemann zeta function.

Contribution

It establishes a comprehensive expression of the Witten multiple zeta function's special values at nonnegative integers in terms of alternating Euler sums, extending understanding of these functions.

Findings

Special values at nonnegative integers are expressible by alternating Euler sums.

Most values of weight w>2 are linear combinations of sums of depth at most two.

Exception occurs when only the last two variables are nonzero, involving ζ(w-1).

Abstract

In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w>2 is a finite rational linear combination of alternating Euler sums of weight w and depth at most two, except when the only nonzero argument is one of the two last variables in which case $\zeta(w-1)$ is needed.