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?



When does a p-adic function have a Mahler expansion?
It is false for the valuation ring in any nontrivial finite extension of $\mathbb{Q}\_p$. The coefficients of the Mahler expansion of a continuous function $\mathcal{O} \to \mathbb{C}\_p$ are determined by its restriction to $\mathbb{Z}\_p$ (they are given as $n$-th differences of the sequence of values on nonnegative integers, in fact). But there are different continuous functions $\mathcal{O} \to \mathbb{C}\_p$ with the same restriction to $\mathbb{Z}\_p$.


Even worse, the Mahler expansions need not even converge because if $x$ is not in $\mathbb{Z}\_p$, the binomial coefficient values may have negative valuation.


**EDIT:** As Kevin Buzzard and dke suggest, one can give a positive answer if your question is interpreted differently. The point of this edit is to make a few explicit remarks in these two directions.


1) If it is known in advance that $f \colon \mathcal{O} \to \mathbb{C}\_p$ is represented by a single convergent power series, then the Mahler expansion of $f|\_{\mathbb{Z}\_p}$ converges to $f$ on all of $\mathcal{O}$. This can be deduced from the theorem that a continuous function $\mathbb{Z}\_p \to \mathbb{C}\_p$ is analytic if and only if the Mahler expansion coefficients $a\_n$ satisfy $a\_n/n! \to 0$ (see Theorem 54.4 in *Ultrametric calculus: an introduction to $p$-adic analysis* by W. H. Schikhof).


2) If one chooses a $\mathbb{Z}\_p$-basis of $\mathcal{O}$, then $f$ can be interpreted as a continuous function $\mathbb{Z}\_p^r \to \mathbb{C}\_p$, and any such function has a multivariable Mahler expansion 
$$\sum a\_n \binom{x\_1}{n\_1} \cdots \binom{x\_r}{n\_r},$$
where the sum is over tuples $n=(n\_1,\ldots,n\_r)$ with $n\_i \in \mathbb{Z}\_{\ge 0}$, and $a\_n \to 0$ $p$-adically.


