Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$ where each $a_i$ is either zero (and must be such for all but finitely many $i<0$) or a root of unity (i.e., a $(q-1)$-th root of unity), collectively known as the Teichmüller representatives. It is now tempting, if $L$ is an arbitrary field containing the $(q-1)$-th roots of unity, to define a (Laurent) power series $f_\xi \in L((t))$ associated to $\xi$ by: $f_\xi = \sum a_i t^i$, assuming we have chosen an isomorphism between the $(q-1)$-th roots of unity in $K$ and those in $L$.
My question is basically whether anything intelligent can be said about $f_\xi$, say, when $\xi$ is a rational, apart from the obvious things like the fact that $f_\xi(p) = \xi$ if we take $L=K$ (with the identity map on roots of unity).
For definiteness, let me concentrate on the first non-trivial case (because already it seems very hard to say anything at all): take $p=5$ and $\xi=2$. Letting $\zeta$ be the $4$-th root of $1$ in $\mathbb{Q}_5$ that is congruent to $2$ mod $5$, we have
$$2 = \zeta - 1 \cdot 5 - \zeta \cdot 5^2 + 0 \cdot 5^3 - 1 \cdot 5^4 + \zeta \cdot 5^5 - 1 \cdot 5^6 - 1 \cdot 5^7 + 1 \cdot 5^8 + \zeta \cdot 5^9 + \zeta \cdot 5^{10} - 1 \cdot 5^{11} + 0 \cdot 5^{12} - \zeta \cdot 5^{13} - 1 \cdot 5^{14} + O(5^{15})$$
and I am asking about the formal series
$$2 + 4 \cdot t + 3 \cdot t^2 + 0 \cdot t^3 + 4 \cdot t^4 + 2 \cdot t^5 + 4 \cdot t^6 + 4 \cdot t^7 + 1 \cdot t^8 + 2 \cdot t^9 + 2 \cdot t^{10} + 4 \cdot t^{11} + 0 \cdot t^{12} + 3 \cdot t^{13} + 4 \cdot t^{14} + \cdots \in \mathbb{F}_5[[t]]$$
(the sequence of coefficients does not appear in the OEIS, which is mildly surprising) or about
$$i - 1 \cdot z - i \cdot z^2 + 0 \cdot z^3 - 1 \cdot z^4 + i \cdot z^5 - 1 \cdot z^6 - 1 \cdot z^7 + 1 \cdot z^8 + i \cdot z^9 + i \cdot z^{10} - 1 \cdot z^{11} + 0 \cdot z^{12} - i \cdot z^{13} - 1 \cdot z^{14} + \cdots \in \mathbb{C}[[z]]$$
Here are some examples of questions which I think are interesting:
is the former algebraic/automatic?
does the latter satisfy some nontrivial differential equation?
can it be extended holomorphically anywhere beyond the unit disk?
does it take interesting values at interesting points?
On a related line, we could consider the $5$-adic quantity
$$- \zeta - 1 \cdot 5 + \zeta \cdot 5^2 + 0 \cdot 5^3 - 1 \cdot 5^4 - \zeta \cdot 5^5 + \cdots$$
obtained by exchanging $\pm\zeta$ in the expansion of $2$ above: in other words, it is the value $f_2(5)$ where $f_2$ has been defined using the unique non-identity automorphism of the $4$-th roots of unity; can anything intelligent be said about that quantity? (e.g., is it rational? experimentally, it doesn't look like it is).
Edit: I should have recalled that, for $\xi$ rational, the sequence of coefficients can be computed from Witt polynomials. Namely, assuming $p$ does not divide the denominator of $\xi$, we let $A_0 = \xi$ and by induction $A_n = (\xi - \sum_{i=0}^{n-1} (A_i)^{p^{n-i}}\cdot p^i)/p^n$: then the $A_i$ are rationals and $p$-adic integers, and $a_i$ is (the root of unity which coincides with) $A_i$ mod $p$. Unfortunately, when seen as integers (if $\xi$ is an integer), the $A_n$ grow extremely rapidly in absolute value, and I don't see how that can be made useful.
