Functional equations for higher logarithms

Abstract

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li _m (z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.