Let $K$ be a field. Are there books or articles discussing completions of $K(x)$ with respect to the metric induced by the $p$-adic valuation $|\;\;|_p$ where $p\in k[x]$ is irreducible and **different from $x$**?

I am interested in the description of the coefficients of the $p$-adic expansion of a rational function in the completion of $(K(x),|\;\;|_p)$.

For example, in the case $p=x$, the coefficients of the $x$-adic expansion of a rational function in $K((x))$ is an eventually linearly recurrent sequence.
A sequence $(r_n)_n$ of elements of $K$ is *linearly recurrent*
if there exist constants $c_1,c_2,\dots,c_k\in K$ such that
\begin{equation}\label{linear recurrence}
r_{n+k}=c_1r_{n}+c_2r_{n+1}+\cdots+c_kr_{n+k-1}
\end{equation}
for all $n\in\mathbb{N}$.
What about the case $p\neq x$?