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

TL;DR

This paper extends the study of values at non-positive integers of multi-variable twisted multiple zeta-functions to cases with general polynomial denominators, providing explicit formulas and revealing some transcendental values.

Contribution

It introduces a method to evaluate these zeta-functions with polynomial denominators at non-positive integers, generalizing previous results for linear and power-sum forms.

Findings

Explicit formulas for values at non-positive integers with polynomial denominators

Reduction to fully twisted case using Mellin-Barnes integral formula

Identification of some transcendental values

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.