ABSTRACT. — The n-th order polylogarithm Ln(z) is defined by the seriesE^Ll zk''t^n on tne Q?611 unit disc. This function has multivalued analytic prolon-gation to C \ {0,1}. The same series ^^ zk f^ defines an analytic p-adic functionon the open unit disc in Cp (a completion of an algebraic closure of Q at some placeabove p). The global p-adic analogues of the functions L\n(z) are constructed in theframe of rigid analysis. These functions are p-adic polylogarithms. In this paper we givesufficient and necessary conditions to have a functional equation of polylogarithms interms of maps induced by rational functions on etale fundamental groups of projectivelines minus finite numbers of points.