Drinfel’d associator and relations of some special functions

Abstract

We observe the differential equation dG(z) / dz = (x0 / z + x1 / (1- z))G(z) in the space of power series of noncommutative indeterminates x0, x1 , where the coefficients of G(z) are holomorphic functions on the simply connected domain ℂ \ [(-∞,0)∪(1,+∞)].Researches on this equation in some conditions turn out different solutions which admit Drinfel'd associator as a bridge. In this paper, we review representations of these solutions by generating series of some special functions such as multiple harmonic sums, multiple polylogarithms and polyzetas. Thereby, relations in explicit forms or asymptotic expansions of these special functions from the bridge equations are deduced by identifying local coordinates.