Functional equations for higher logarithms, by Herbert Gangl

The author has submitted an updated version, July 26, 2002.

Following earlier work by Abel and others, Kummer gave functional equations for the polylogarithm function Li_m(z) up to m=5 in 1850, 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.

Herbert Gangl <>