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 \[d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad (x,y\in \R,\,k\in\{0,\ldots,n\}) \] is studied, where $\Delta_n:=\big\{(i,j)\in\Z\times\Z\mid 0\leq i,j\mbox{and}i+j\leq n\big\}$ and $\Gamma\colon\Delta_n\to\R$ is a symmetric function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n}\colon \R\to\R$ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon\R\to\R$, 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 $\R^{n+1}$.