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

The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations $$d_k(xy)=\sum _{i=0}^k\mathrm{\Gamma }(i,k-i)d_i(x)d_{k-i}(y)(x,y\in \mathbb{R},k\in \{0,\dots ,n\})$$ is studied, where ${\mathrm{\Delta }}_n:=\{(i,j)\in \mathbb{Z}\times \mathbb{Z}\mid 0\le i,jandi+j\le n\}$ and $\mathrm{\Gamma }\text{\hspace{0.17em}⁣}:{\mathrm{\Delta }}_n\to \mathbb{R}$ is a symmetric function such that $\mathrm{\Gamma }(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear dependence and independence of the additive solutions $d_0,d_1,\dots ,d_n\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$, the iterates $d^0,d^1,\dots ,d^n$ of $d$ are shown to be linearly independent, and the graph of the mapping $x\mapsto (x,d^1(x),\dots ,d^n(x))$ to be dense in ${\mathbb{R}}^{n+1}$.