Identities for Generalized Polylogarithms

E. A. Ulanskii

Matematičeskie zametki, Tome 73 (2003) no. 4, pp. 613-624

Résumé

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 $\mathbb C(z)$.