Identities for Generalized Polylogarithms

We study the behavior of generalized polylogarithms under the action of the group of fractional-linear transformations of the argument. This group is formed by the transformations $z\mapsto1-z$ and $z\mapsto-z/(1-z)$, the last of which allows us to obtain identities of the form
$$ \operatorname{Li}_k\biggl(\frac{-z}{1-z}\biggr) =-\sum_{|\bar s|=k}\operatorname{Li}_{\bar s}(z). $$
We prove that these identities imply the linear independence of generalized polylogarithms and the algebraic independence of classical polylogarithms over the field $\mathbb C(z)$.