Stirling polynomials and multiple zeta (star) functions at non-positive integers

TL;DR

This paper provides explicit formulas for multiple zeta functions and star functions at non-positive integers using Stirling polynomials, revealing new connections and generalizations in special number theory functions.

Contribution

It introduces explicit formulas for MZFs and MZSFs at non-positive integers using Stirling polynomials, and explores their interrelations and links to Gregory coefficients.

Findings

Explicit formulas for MZFs and MZSFs at non-positive integers.

Connections between MZFs, MZSFs, and Gregory coefficients.

New insights into the structure of multiple zeta functions.

Abstract

It is known that the values of multiple zeta functions (MZFs) at non-positive integers can be expressed by Bernoulli numbers. This paper gives explicit formulas for the values of MZFs and multiple zeta star functions (MZSFs) at non-positive integers using Stirling polynomials. We also study the following two points: a connection between MZFs and MZSFs at non-positive integers and a connection between reverse values and generalized Gregory coefficients studied by Matsusaka, Murahara, and Onozuka.