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

On the functional equation $\displaystyle \alpha\bf{u}+\mathcal{C}\star(\chi \bf{u})=\bf{f}$