Interpolation with $p$-adic formal power series in $\mathbb Z_p[[x]]$ It's a classical exercise in complex analysis that one could find a holomorphic function with given values on a set of points on the complex plane without limit points.
What about the p-adic analogue? For example given $a_n \in \Bbb N$ for any positive integer $n$, when could we find $f \in \mathbb Z_p[[x]]$ such that $f((1+p)^n-1)=a_n$ for all $n$?
 A: I think your choice of the set $\{(1+p)^n-1\}$ on which you want $f$ to take the values $a_n$ is not so good.
Except when $p=2$, the logarithm, defined on the multiplicative group $1+p\Bbb Z_p$ by the usual formula $\log(1+x)=-\sum_1^\infty(-x)^m/m$, and the exponential $\exp(x)=\sum_0^\infty x^m/m!\,$, are convergent on the closed subdisk $p\Bbb Z_p$ and are inverse to each other. Once you make the associate analytic change of coordinatization, therefore, you’re really asking to find an analytic $F$ such that for every $n\ge0$, you get $F(np)=a_n$. You’re really asking whether $pn\mapsto a_n$ is a $p$-adically analytic function of $n$. (The $2$-adic situation, where you’re asking about powers of $3$, is different only in detail.)
As I said in my comment, the roots of unity are wide-spaced in the $p$-adic universe, unlike the complex universe, where they’re dense in a subset that’s locally $\Bbb R$; the reverse is true of the natural integers, or your numbers $(1+p)^n-1$, which are wide-spaced in the complex universe, but dense in a subset that’s locally $\Bbb Q_p$.
