Values at non-positive integers of partially twisted multiple zeta-functions II

2025/06/25 by Driss Essouabri, Essouabri, Driss, Kohji Matsumoto +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2506.20150

Abstract

We study the values at non-positive integer points of multi-variable twisted multiple zeta-functions, whose each factor of the denominator is given by polynomials. The fully twisted case was already answered by de Crisenoy. On the partially twisted case, in one of our former article we studied the case when each factor of the denominator is given by linear forms or power-sum forms. In the present paper we treat the case of general polynomial denominators, and obtain explicit forms of the values at non-positive integer points. Our strategy is to reduce to the theorem of de Crisenoy for the fully twisted case, via the multiple Mellin-Barnes integral formula. We observe that in some cases the obtained values are transcendental.

Citations

• Values and derivative values at nonpositive integers of generalized multiple Hurwitz zeta functions
• Asymptotic behavior of the Hurwitz-Lerch multiple zeta function at non-positive integer points
• Values of multiple zeta-functions with polynomial denominators at non-positive integers
• Multiple Stieltjes constants and Laurent type expansion of the multiple\n zeta functions at integer points
• Values at non-positive integers of partially twisted multiple zeta-functions I
• Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions
• Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points
• Functional equations for double series of Euler-Hurwitz-Barnes type with coefficients
• Functional equations for double series of Euler type with coefficients
• Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers

Related

• Values at non-positive integers of partially twisted multiple zeta-functions I
• Values of multiple zeta-functions with polynomial denominators at non-positive integers
• Relations between values at T-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables
• Some explicit values of a q-multiple zeta function whose denominator power is not uniform
• Multiple zeta functions at regular integer points