Hyperlogarithms: Functions on Free Monoids
Vincel Hoang Ngoc Minh
TL;DR
The work develops an algebraic framework for representing functions on the free monoid generated by an alphabet, casting their graphs as noncommutative generating series and exploiting two complementary Hopf-algebra structures (concatenation and shuffle with a $\phi$-deformation). It delivers explicit diagonal factorization via Lyndon-based bases, and proves that rational (representative) series correspond precisely to finite-rank linear representations, with Sweedler duals of the corresponding bialgebras recovering the algebras of representative series when the base ring is a field $K$. The constructions unify classical objects like polylogarithms and harmonic sums with modern noncommutative generating-series methods, and extend to the $\phi$-shuffle setting, providing explicit criteria for rationality and a robust factorization/decomposition toolkit (Lazard factorization) for these series. The results have potential impact on the study of iterated integrals and differential systems in the free-monoid context, including hyperlogarithms and Fliess-type representations of linear dynamical systems.
Abstract
To factorize and to decompose the graphs of representative functions on the free monoid X * (generated by the alphabet X ) with values in the ring A containing Q, we examine various products of series (as concatenation, shuffle and its $φ$ -deformations) and co-products, which are such that their associated non graded bialgebras are isomorphic, for A is a field K, to the Sweedler's dual of the graded noncommutative co-commutative K-bialgebra of polynomials.