Representations of skew linear recurrent sequences of maximal period over finite field

Let $p$ be a prime number, $R=\mathrm{GF}(q)$ be a finite field, where $q = p^r$, $S=\mathrm{GF}(q^{n})$ be its extension of degree $n$ and $\check{S}$ be a ring of linear transforms of the vector space ${}_RS$. A sequence $v$ over $S$ with a recursion law of the form
$$\forall i\in\mathbb{N}_0 \colon v(i+m)= \psi_{m-1}(v(i+m-1))+\ldots+\psi_0(v(i)), \psi_0,\ldots,\psi_{m-1 }\in\check{S},$$
is called skew linear recurrent sequence over $S$ of order $m$ with the characteristic polynomial $\Psi(x) = x^m - \sum_{j=0}^{m-1}\psi_jx^j$. It is well known that maximal period of such sequence is equal to $q^{mn}-1$. Let $v$ be a skew LRS of maximal period over $S$, $J$ be an arbitrary ring with identity $\mathbf{e}$ such that $q\mathbf{e}$ is not a zero divisor and $f: S \to J$ be a map. Below under certain conditions we describe the annihilator of the sequence $f(v)$.