Basing on Picard-Vessiot theory of noncommutative differential equations and
algebraic combinatorics on noncommutative formal series with holomorphic
coefficients, various recursive constructions of sequences of grouplike series
converging to solutions of universal differential equation are proposed. Basing
on monoidal factorizations, these constructions intensively use diagonal series
and various pairs of bases in duality, in concatenation-shuffle bialgebra and
in a Loday's generalized bialgebra. As applications, the unique solution,
satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is
provided by d\'evissage