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For each multi–index
of positive integers, one defines the generalized *polylogarithms*

(1) |

This series in converges at the interior of the open unit disk. In , these polylogarithms yield the generalized Riemann function

(2) |

which converges for .

Let be the alphabet on two letters and . Any multi–index can be encoded by a unique word

(3) |

Now each function , which is
also denoted by , can be
obtained by an *iterated integral* as follows:

and

(4) |

for any . These integrals are functions defined on the universal Riemann surface above . The real number is also denoted by for all .

It is useful to extend the above definition of to the case when . For each , we take

(5) |

and we extend the definition to using (4). These generalized polylogarithms are again defined on and we will prove the important fact that

(6) |

is a Lie exponential for all .

The monodromy of the *classical* polylogarithms , when turning around the point has been computed previously

(7) |

From a theoretical point the monodromy of the series can be computed using tools developed by J. Écalle. Notice that the monodromy of in particular yields the monodromy of each for .

In this paper, we give an explicit method to compute the monodromy. Our
algorithm has been implemented in the

**Coauthors:** H.N. Minh and M. Petitot

**Occasions:** ISSAC 1998, Rostock, August 20

**Documents:** slides