The aim of this paper is to propose a method of analytic continuation of the Carlitz
multiple (star) polylogarithms to the whole space by using Artin–Schreier equations
and present a treatment of their branches by introducing the notion of monodromy
modules. As applications of this method, we obtain (1) a method of continuation of
the logarithms of higher tensor powers of the Carlitz module, (2) the orthogonal
property (Chang–Mishiba functional relations), (3) a branch independency of the
Eulerian property.
