Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this?

$\begin{align*} f_{n,k}(x) & =a(x)f_{n-1,k}(x)+b(x)f_{n-2,k}(x) & n\equiv1\,\mbox{(mod k)}\\ & =f_{n-1,k}(x)+a(x)b(x)f_{n-2,k}(x) & n\equiv2\,\mbox{(mod k)}\\ & =f_{n-1,k}(x)+b(x)f_{n-2,k}(x) & \mbox{o.w.}\end{align*}$

My preliminary work suggests that for all $k$, it should collapse down into a single second-order recurrence relation:

$\begin{align*} f_{n,k} & =a_k(x)f_{n-k,k}+b_k(x)f_{n-2k,k}(x)\end{align*}$

with the original recurrence divided into $k$ separate solutions each defined by initial conditions $f_{j,k}(x)$ and $f_{j+k,k}(x)$, $0\leq j < k$, where the functions $a_k(x)$ and $b_k(x)$ are themselves defined by a specific pair of recurrence relations in $k$. But the modular structure's giving me troubles in proving either the collapse or, given the collapse, the validity of the recurrences potentially defining $a_k(x)$ and $b_k(x)$ through any sort of inductive approach, and I haven't had any luck digging up any references to a recurrence structure of even a vaguely similar form. Even just something similar in nature could give me a lead into pinning this down.