Additive solvability and linear independence of the solutions of a system of functional equations

Abstract

The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations

$$d_k(xy) = \sum_{i=0}^k \Gamma (i, k - i)d_i(x)d_{k-i}(y) \; \; \; (x, y \in \mathbb{R}, k \in \{0,...., n\}$$

is studied, where Δ_n:= {(i, j) ∈ ℤ × ℤ | 0 ≤ i, j and i + j n } and Γ: Δ_n → ℝ is a symmetric function such that Γ (i, j) = 1 whenever i · j = 0. On the other hand, the linear dependence and independence of the additive solutions d₀, d₁,..., d_n: ℝ → ℝ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation d: ℝ → ℝ, the iterates d⁰, d¹,..., dⁿ of d are shown to be linearly independent, and the graph of the mapping x ↦ (x, d¹(x),..., dⁿ(x)) to be dense in ℝⁿ⁺¹.