  We study in this paper the functional equation $\displaystyle \alpha
\mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t)$ where
$\alpha\in\mathbb{C}^{d\times d}$,
$\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being
unknown. The term $\mathcal{C}\star(\chi \mathbf{u})(t)$ denotes the discrete
convolution of an almost zero matricial mapping $\mathcal{C}$ with discrete
support together with the product of $\mathbf{u}$ and the characteristic
function $\chi$ of a fixed segment. This equation combines some aspects of
recurrence equations and/or delayed functional equations, so that we may
construct a matricial based framework to solve it. We investigate existence,
unicity and determination of the solution to this equation. In order to do
this, we use some new results about linear independency of monomial words in
matrix algebras.
