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 [Formula: see text] (noncommutative symmetric) system [28] ΩFthas been constructed. Consequently, we get a Hopf algebra homomorphism [Formula: see text] from the Hopf algebra [Formula: see text] [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 [Formula: see text] system ΩFtby using some identities of NCSFs derived in [9] and the homomorphism [Formula: see text]. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system ΩFtfor the Taylor series expansions of u(Ft) and [Formula: see text]; the D-Log and the formal flow of Ftand inversion formulas for the inverse map of Ft. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.