Some explicit values of a $q$-multiple zeta function at roots of unity

TL;DR

This paper provides explicit formulas for a specific $q$-multiple zeta function at roots of unity, connecting it to Stirling numbers and Bell polynomials, advancing understanding of these special values.

Contribution

It introduces explicit formulas for $q$-multiple zeta functions at roots of unity using combinatorial tools like Bell polynomials and Stirling numbers.

Findings

Explicit formulas for $q$-multiple zeta values at roots of unity.

Connections between zeta values and Stirling numbers.

Use of determinants and Bell polynomials in deriving formulas.

Abstract

In this paper, we give the values of a certain kind of $q$-multiple zeta functions at roots of unity. Various multiple zeta values have been proposed and studied by many researchers, but these multiple zeta values naturally arise from generalizations of Stirling numbers. It is interesting, but by no means easy, to show the values explicitly in certain cases. We give explicit formulas by using Bell polynomials, determinants, $r$-Stirling numbers, etc.