On the solutions of universal differential equation

Abstract

The objective of this work is to give expressions for solutions of universal differential equation and finally, as application, provides the unique grouplike solution, by dévissage, of Knizhnik-Zamolodchikov equation satisfying asymptotic conditions, i.e. solution of KZn can be obtained using solution of KZ n−1 and the noncommutative generating series of hyperlogarithms. These expressions are obtained using the algebraic combinatorics on noncommutative formal series with holomorphic coefficients and a Picard-Vessiot theory of noncommutative differential equations.