Let $\mathbb{Z}\_p$ be the ring of $p$-adic integers, $\mathbb{Q}\_p$ the field of fractions of $\mathbb{Z}\_p$, and $\mathbb{C}\_p$ the completion of the algebraic closure of $\mathbb{Q}\_p$. Let $v\_p$ be the $p$-adic valuation on $\mathbb{C}\_p$ with $v\_p(p)=1$. For each $m\ge1$ define the set
$$
A\_m=\mathbb{C}\_p\setminus\left(
\bigcup\_{a=0}^{p^m-1}
\left\{x\in\mathbb{C}\_p:v\_p(x-a)\ge m\right\}
\right).
$$ 
Then we can write $\mathbb{C}\_p\setminus\mathbb{Z}\_p=\cup\_{m\ge1}A\_m$.


A *Krasner analytic function* $f$ on $\mathbb{C}\_p\setminus\mathbb{Z}\_p$ is a function $f:\mathbb{C}\_p\setminus\mathbb{Z}\_p\to\mathbb{C}\_p$ such that, for each $m\ge1$, $f$ restricted to $A\_m$ is a uniform limit of a sequence of rational functions with poles outside $A\_m$. In other words, for all $m\ge1$, $f$ is an *analytic element* on $A\_m$. This makes sense because each $A\_m$ is a quasi-connected subset of $\mathbb{C}\_p$.


**Question:** Is there any known characterization of Krasner analytic functions on $\mathbb{C}\_p\setminus\mathbb{Z}\_p$? 


By a characterization I mean something like the Amice-Fresnel theorem (see Alain Robert's book on $p$-adic analysis, page 348).


My main motivation is that the second derivative of Diamond's $p$-adic log-gamma function is an analytic element on $\mathbb{C}\_p\setminus\mathbb{Z}\_p$ (see Jack Diamond's 1977 paper, Theorem 12), and I was wondering if this is a general phenomenon for some "nice" functions defined by means of the Volkenborn integral (Kubota-Leopoldt sums), as is the case of Diamond's $p$-adic log-gamma function.



Characterization of Krasner analytic functions on the complement of $p$-adic integers
If you impose some growth conditions on the boundary of ${\mathbb C}\_p\setminus {\mathbb Z\_p}$ to the Krasner-analytic function $F$, plus some invariance under the absolute Galois group of ${\mathbb Q}$, you have a caracterization in the spirit of Amice-Fresnel theorem. I use $p$-adic absolute value instead of $p$-adic valuation. 


For example if you impose the conditions


1. Suppose that $\sup\_{x\in A\_m}\vert F(x)\vert\_p\leq p^{m}$ for all $m\in {\mathbb N}$
2. Suppose that the Krasner-analytic function $F$ is ${\rm Gal\_{\text{cont}}(\mathbb C}\_p/{\mathbb Q}\_p)$-equivariant, that is for all $\sigma\in {\rm Gal\_{\text{cont}}(\mathbb C}\_p/{\mathbb Q}\_p)$ and for all $x\in {\mathbb C}\_p\setminus {\mathbb Z\_p}$ one has $F(\sigma(x))=\sigma(F(x))$.
3. Suppose that $\lim\_{\vert x \vert\to \infty}\vert F(x) \vert=0$


Then there exists a ${\mathbb Z}\_p$-valued measure, $\mu$, on ${\mathbb Z}\_p$ such that:
\begin{equation\*}
F(x)= \int\_{t\in{\mathbb Z}\_p}\frac{d\mu(t)}{t-x}
\end{equation\*}
(see the references below).


Let $e\_n^\*$ be the measure on ${\mathbb Z}\_p$ such that $\int\_t\binom{t}{m}de\_n^\*=\delta\_m^n$ ${\scr C}({\mathbb Z}\_p,{\mathbb Z}\_p)$. It is known that any ${\mathbb Z}\_p$-valued measure $\mu$ on ${\mathbb Z}\_p$ can be written
\begin{equation\*}
\mu=\sum\_{n\geq 0} b\_ne\_n^\*, \ b\_n \in {\mathbb Z}\_p\, .
\end{equation\*}
From this one get easily, using Mahler's expansion of the $p$-adic continuous function $t\mapsto \frac{1}{t-x}$, that the Krasner-analytic function $F$ subject to conditions 1,2,3 has the following expansion on ${\mathbb C}\_p\setminus {\mathbb Z\_p}$
\begin{equation\*}
F(x)=\sum\_{n\geq 0} b\_n\sum\_{k=0}^n(-1)^{k}\binom{n}{k}\frac{1}{k-x}= \sum\_{n\geq 0} \frac{b\_nn!}{x(x-1)\dots(x-n)}
\end{equation\*}


This result can be extended in many directions.


(references:


D. Barsky, Transformation de Cauchy $p$-adique et algèbre d'Iwasawa, *Math. Ann. **232** (1978), 255-266*,


or 


V. Alexandru, C.C. Niţu, M. Vâjâitu, A. Zaharescu, On the norm of Krasner analytic functions with applications to transcendence results, *Journal of Pure and Applied Algebra **219** (2015) 4607–4618*)


