A NOTE ON FUNCTIONAL-EQUATIONS OF THE P-ADIC POLYLOGARITHMS

The n-th order polylogarithm L(n)(z) is defined by the series SIGMA(k=1)infinity z(k)/k(n) on the open unit disc. This function has multivalued analytic prolongation to C \ {0, 1}. The same series SIGMA(k=1)infinity z(k)/k(n) defines an analytic p-adic function on the open unit disc in C(p) (a completion of an algebraic closure of Q at some place above p). The global p-adic analogues of the functions Li(n)(z) are constructed in the frame of rigid analysis. These functions are p-adic polylogarithms. In this paper we give sufficient and necessary conditions to have a functional equation of polylogarithms in terms of maps induced by rational functions on etale fundamental groups of projective lines minus finite numbers of points.