Abstract. The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \Eq{*}{ d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad(x,y\in{\msbm R}, k\in\{0,\ldots,n\}) } is studied, where $\Delta_n:=\big\{(i,j)\in\Z \times\Z \mid0\leq i,j\mbox{ and }i+j\leq n\big\}$ and $\Gamma\colon \Delta_n\to{\msbm 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{\msbm R} \to{\msbm R} $ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon{\msbm R} \to{\msbm 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 ${\msbm R} ^{n+1}$.

DOI: 10.14232/actasm-014-534-6

AMS Subject Classification (1991): 16W25, 39B50

Keyword(s): derivation, higher order derivation, iterates, linear dependence

Received April 23, 2014, and in revised form August 19, 2014. (Registered under 34/2014.)