Universit\’e des Sciences et Technologies de Lille

$\bullet $ The monodromy’s study of Fuchsian hypergeometric differential equation provides a natural framework for the explicit determination of rational approximations of polylogarithmic functions.Thus, we can obtain almost without calculation explicit determination of many polynomials and hypergeometric power series related to their Pad\’e approximations, From now on, using a classical way, one can study the arithmetic nature of numbers related to the values taken by these functions. It is an expanded version of the conference given at the sympo-sium on New aspects of analytic Number theory, held at the RIMS of the university of Kyoto in october. 27-29, 2008. I would like to thank the organizer Professor Takao Komatsu and also N.Hirata Kohno for their invitation to come to Japan.