On the linear relations among parametrized multiple series

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 parametrized multiple series. Specifically, we prove the following two statements: the linear part of the Kawashima relation for multiple zeta values can be generalized to 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.