Identities for generalized polylogarithms. (English. Russian original) Zbl 1093.11049

Ulanskiĭ, E. A.

Math. Notes 73, No. 4, 571-581 (2003); translation from Mat. Zametki 73, No. 4, 613-624 (2003).

Summary: 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 \(z\mapsto 1-z\) and \(z\mapsto-z/(1- z)\), the last of which allows us to obtain identities of the form \(\operatorname{Li}_k\big(\frac{-z}{1-z}\big) =-\sum_{|\bar s| =k} \operatorname{Li}_{\bar s}(z)\). We prove that these identities imply the linear independence of generalized polylogarithms and the algebraic independence of classical polylogarithms over the field \(\mathbb C(z)\).

MSC:

11G55	Polylogarithms and relations with \(K\)-theory
11J81	Transcendence (general theory)
11M32	Multiple Dirichlet series and zeta functions and multizeta values
33B30	Higher logarithm functions
33C80	Connections of hypergeometric functions with groups and algebras, and related topics

Keywords:

generalized polylogarithm; fractional-linear transformation; multiple zeta function; classical polylogarithm; algebraic independence