Narrow Results

Let K be any unital commutative ℚ-algebra and z = (z₁, z₂, …, z_n) 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 F_t(z) = z - H_t(z) of K[[t]]〈〈z〉〉 with H_t=0(z) = 0 and o(H(z)) ≥ 1, a (noncommutative symmetric) system [28] Ω_Ft 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 Ω_Ft 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 Ω_Ft for the Taylor series expansions of u(F_t) and ; the D-Log and the formal flow of F_t and inversion formulas for the inverse map of F_t. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.