Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with

$$ f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}. $$

This is known as the Mahler expansion of $f$. Here, to make sense of $x \choose n$ for $x \not\in \mathbb{Z}$ define

$$ {x\choose n} = \frac{x(x-1)\ldots(x-n+1)}{n!}. $$

There's an important application to number theory: expressing a function in terms of its Mahler expansion is one step in translating the old fashioned interpolation-based language of $p$-adic $L$-functions into the more modern measure-theoretic language. This application is explained in Coates and Sujatha's book *Cyclotomic Fields and Zeta Values*.

However, when the $p$-adic $L$-function is more complicated than Kubota-Leopoldt's, it seems to me that this "translation" really requires one to be able to write down a Mahler expansion of a function $f$ with much larger domain, e.g. the ring of integers of the completion of the maximal unramified extension of $\mathbb{Q}_p$ or some finitely ramified extension. (See, for instance, line (8), p. 19 of de Shalit's *Iwasawa Theory of Elliptic Curves with Complex Multiplication*).

I can't find a published reference that these functions really have Mahler expansions. Mahler's paper uses in an essential way that the positive integers are dense in $\mathbb{Z}_p$, so it doesn't instantly generalize.

So is it true or false that for a ring of integers $\mathcal{O}$ in a finitely ramified complete extension of $\mathbb{Q}_p$, and a function $f: \mathcal{O} \rightarrow \mathbb{C}_p$, there is a Mahler expansion as above?