On a multi-variable Arakawa-Kaneko zeta function for non-positive or positive indices

Abstract: The Arakawa-Kaneko zeta function (the xi function) and Kaneko-Tsumura zeta function (the eta function) are defined as the Mellin transformation of the generating function of multi-poly-Bernoulli numbers and notably related to multi-poly-Bernoulli numbers and multiple zeta values. One striking discovery is the duality of the multi-variable eta function. Specifically, one can obtain the duality formula among multi-indexed poly-Bernoulli numbers of B-type and, using the formula for special values of the multi-variable eta function in terms of a linear combination of multiple zeta values, a new family of relations among multiple zeta values. In this talk, we introduce our study on the multi-variable xi function. First, its analytic continuation to an entire function. Second, a duality formula among multi-indexed poly-Bernoulli numbers of C-type which is regarded as a special case of the possible duality of the multi-variable xi function. Third, an explicit procedure for writing the special values of the multi-variable xi function as a linear combination of multiple zeta values.

number theory

Audience: researchers in the topic

Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October