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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. 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