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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 $p\neq x$?

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    $\begingroup$ Here is an example to think about: completing $\mathbf Q(x)$ at $|\cdot|_{x^2+1}$ will be $\mathbf Q(i)((\pi))$ where $\pi = x^2+1$. This is analogous to completing a number field at a prime ideal with residue field degree greater than $1$. Completing $K(x)$ at $x-c$ gives you $K((x-c))$, so not interesting for your purposes. $\endgroup$
    – KConrad
    Commented May 2, 2017 at 8:23

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