  We investigate a connection between the differential of polylogarithms (as
considered by Cathelineau) and a finite variant of them. This allows to answer
a question raised by Kontsevich concerning the construction of functional
equations for the finite analogs, using in part the p-adic version of
polylogarithms and recent work of Besser.
  Kontsevich's original unpublished note is supplied (with his kind permission)
in an "Appendix" at the end of the paper.
