On the linear relations among parametrized multiple series

TL;DR

This paper fully characterizes the linear relations among parametrized multiple series, extending known relations for multiple zeta values and showing all such relations derive from the Kawashima relation.

Contribution

It generalizes the linear relations for multiple zeta values to parametrized series and proves these relations are generated by the linear part of the Kawashima relation.

Findings

Linear part of Kawashima relation extends to parametrized series

All linear relations among parametrized series are derived from Kawashima relation

Complete characterization of linear relations among parametrized multiple series

Abstract

Parametrized multiple series are generalizations of the multiple zeta values introduced by Igarashi. In this work, we completely determine all the linear relations among these parameterized multiple series. Specifically, we prove the following two statements: the linear part of the Kawashima relation for multiple zeta values can be generalized to the parametrized multiple series; any linear relations among the parametrized multiple series can be written as a linear combination of the linear part of the Kawashima relation.