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

2025/08/18 by Ishii, Yoni, Takeshi Shinohara, Shinohara, Takeshi
Chemical Engineering · Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT) #Thermodynamic properties of mixtures

paper · pdf · doi:10.48550/arxiv.2508.12577

openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients
• Asymptotic behavior of the Hurwitz-Lerch multiple zeta function at non-positive integer points
• Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers
• Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions
• Desingularization of complex multiple zeta-functions
• Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers

Related

• Some explicit values of a q-multiple zeta function at roots of unity
• A Note on Harmonic number identities, Stirling series and multiple zeta values
• Instant Multiple Zeta Values at Non-Positive Integers and Bernoulli Functions
• Values of multiple zeta-functions with polynomial denominators at non-positive integers
• Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients