Alternating variants of multiple poly-Bernoulli numbers and finite multiple zeta values in characteristic 0 and p

TL;DR

This paper explores alternating variants of multiple poly-Bernoulli numbers and finite multiple zeta values in both characteristic 0 and p, providing explicit formulas and analogues in positive characteristic.

Contribution

It introduces alternating extensions of multiple poly-Bernoulli numbers and finite multiple zeta values in characteristic p, expanding the theoretical framework and explicit representations.

Findings

Explicit presentations of alternating finite multiple zeta values in characteristic 0.

Introduction of positive characteristic analogues of alternating finite multiple zeta values.

Any finite multiple zeta value with integer index can be expressed as a linear combination of FMZVs with positive indices.

Abstract

Alternating variants of multiple poly-Bernoulli numbers and finite multiple zeta values in characteristic 0 and p This paper consists of two parts: the characteristic 0 part and the characteristic p part. In characteristic 0 part, we introduce an alternating extension of multiple poly-Bernoulli numbers of K. Imatomi, M. Kaneko and E. Takeda and obtain explicit presentations of the alternating finite multiple zeta values introduced by J. Zhao in term of the alternating extension of multiple poly-Bernoulli numbers. In characteristic p part, we introduce positive characteristic analogues of alternating finite multiple zeta values and express them as special values of finite Carlitz multiple polylogarithms defined by C.-Y. Chang and Y. Mishiba. We introduce alternating variants of R. Harada's multiple poly-Bernoulli-Carlitz numbers, which are analogues of multiple poly-Bernoulli numbers, to obtain explicit presentations of the finite alternating multiple zeta values. We show that any finite multiple zeta value with integer index is expressed as k-linear combination of FMZV's with all-positive indices.