NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND THE INVERSION PROBLEM
Abstract
Let K be any unital commutative ℚ-algebra and z = (z1, z2, …, zn) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]〈〈z〉〉 the formal power series algebra of z over K[[t]]. In [29], for each automorphism Ft(z) = z - Ht(z) of K[[t]]〈〈z〉〉 with Ht=0(z) = 0 and o(H(z)) ≥ 1, a 
(noncommutative symmetric) system [28] ΩFt has been constructed. Consequently, we get a Hopf algebra homomorphism 
from the Hopf algebra 
[9] of NCSFs (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the 
system ΩFt by using some identities of NCSFs derived in [9] and the homomorphism 
. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system ΩFt for the Taylor series expansions of u(Ft) and 
; the D-Log and the formal flow of Ft and inversion formulas for the inverse map of Ft. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.
