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

Takao Komatsu

TL;DR

The paper studies explicit values of a finite $q$-multiple zeta function at roots of unity by introducing $q$-generalized $(r,s)$-Stirling numbers and connecting them to Bell polynomials and determinant expressions. It develops a framework that expresses $Z_n( abla_n;m,s)$ in terms of $Z_n( abla_n;1,s)$ via Bell polynomials and determinants, with an inversion formula linking these representations. Concrete results include explicit formulas for $s=2$ and $s=3$, and connections to degenerate Bernoulli numbers and $r$-Stirling numbers, enabling closed-form evaluations and structural insight. The work provides computable, structurally rich representations for finite $q$-MZVs at roots of unity, highlighting deep ties to classical combinatorial objects and suggesting avenues for extending the results to higher levels.

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.

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

TL;DR

The paper studies explicit values of a finite -multiple zeta function at roots of unity by introducing -generalized -Stirling numbers and connecting them to Bell polynomials and determinant expressions. It develops a framework that expresses in terms of via Bell polynomials and determinants, with an inversion formula linking these representations. Concrete results include explicit formulas for and , and connections to degenerate Bernoulli numbers and -Stirling numbers, enabling closed-form evaluations and structural insight. The work provides computable, structurally rich representations for finite -MZVs at roots of unity, highlighting deep ties to classical combinatorial objects and suggesting avenues for extending the results to higher levels.

Abstract

In this paper, we give the values of a certain kind of -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, -Stirling numbers, etc.
Paper Structure (11 sections, 13 theorems, 70 equations)

This paper contains 11 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. $q$-generalized $(r,s)$-Stirling numbers
  3. First kind
  4. Second kind
  5. Main relations
  6. Proof of the results
  7. Relations with another finite multiple zeta function
  8. The case $s=2$
  9. A different expression
  10. The case $s=3$
  11. Further works

Key Result

Lemma 1

where $\delta_{n,m}$ is the Kronecker delta.

Theorems & Definitions (19)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 5
  • proof
  • ...and 9 more