A family of polynomials $\{P_i(x),i=0,1,\ldots,m\}$ $(m\in\mathcal N_0)$ of degree i is a convolution one if satisfies the functional equation \begin{equation} um^m_{i=0}P_{m-i}(x)P_i(y)=P_m(x+y), \end{equation} for every $x,y\in\mathcal R$. The generalization of (1) is the functional equation \begin{equation} um^m_{i=0}P_{m-i,i}(a,p,q;x,y)=P_m(a,p+q;x+y), \end{equation} where $P_{j,k}(a,p,q;x,y)$ is polynomial of degree $j+k=m$ in two variables, $x$ and $y$, and $a,p,q$ are real parameters. The $n$-dimensional generalization of (1) is \begin{multline} um_{m_1+\cdots+m_n=m}P_{m1,...,m_n}(a,p_1,...,p_n;x_1,...,x_n) =P_m(a,p_1+\cdots+p_n;x_1+\cdots+x_n). \end{multline}