Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions

TL;DR

This paper derives explicit formulas for generalized Euler-Zagier multiple zeta-functions at non-positive integers, expanding understanding of their special values through analytic continuation and an extended Raabe's lemma.

Contribution

It introduces new explicit formulas for multiple zeta values at non-positive integers and extends Raabe's lemma to facilitate these results.

Findings

Explicit formulas for multiple zeta values at non-positive integers

Extension of Raabe's lemma for analytic continuation

Method applicable to a broad class of zeta-functions

Abstract

We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions and then use the analyticity of the values on the parameters defining the multiple zeta-functions to deduce the formulas in the general case. Also, for our aim we prove an extension of "Raabe's lemma" due to E. Friedman and A. Pereira.