Stokes factors and multilogarithms

Let G be a complex, affine algebraic group and a meromorphic connection on the trivial G -bundle over , with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map taking the residue of at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of , and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.