Characterization of Krasner analytic functions on the complement of $p$-adic integers

Question

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.

Answer 1

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)