On the self-similarity of rational power series with matrix coefficients

Let $p$ be a prime, let $d\ge 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P,Q\in A[x_1,\dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that $Q$ is invertible in $A[\text{\hspace{0.17em}⁣}[x_1,\dots ,x_n]\text{\hspace{0.17em}⁣}]$. Let also $\mathcal{M}\text{\hspace{0.17em}⁣}:{\mathbf{Z}}^n\to A$ be the map associating to the $n$-tuple of integers $({\alpha }_1,\dots ,{\alpha }_n)$ the coefficient of the monomial $x_1^{{\alpha }_1}\dots x_n^{{\alpha }_n}$ in the development of the rational fraction $P{Q}^{-1}$ as a power series (the support of $\mathcal{M}$ is contained in ${\mathbf{N}}^n$). Our main result ensures that the map $\mathcal{M}$, viewed as a tiling of ${\mathbf{R}}^n$ by unit cubes with color set $A$, is self-similar. The self-similarity is expressed in terms of invariance under substitutions. By specializing to $d=1$, $n=2$, $P=1$ and $Q=1-x_1-x_2$, we recover the well-known self-similarity feature of the binomial coefficients modulo $p$.