Non commutative generating series, in control theory and special functions

Gerard Jacob (Univ.Lille I, France)

The noncommutative power series appear as a combinatorial tool that is useful
in control theory - where they play the rule of the transfer functions, in a
non linear setting - as well as in the domain of the special functions.  In
these two fields, we introduce them by encoding iterated integrals on some
differential forms, as defined by K. T. Chen.  We use them jointly with free
Lie algebras, Poincarre-Birkhoff-Witt theorem, and Hopf algebras.  Recently, we
obtained new results by use of the noncommutative rational series.  We get so a
good combinatorial preparation for writing down some specific packages in
Computer Algebra Systems.

First we give a short presentation of their use in an identification algorithm
for an input/output  system given as a blackbox behaviour.  Further, we sketch
the results we have obtained in the theory of polylogarithm functions:
monodromy, functional relations, Euler-Zagier sums (or "Multiple Zeta Values"),
Drinfeld associator, renormalization, (and coloured versions by adjonction of
roots of unity), Hurwitz polylogarithm functions, and computation of the table
of the MZV's.