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Let $f: \mathbb Z_{\ge 0} \to \mathbb C_p$ be any function. My understanding is that Mahler's theorem says that $f$ extends to a continuous function $f: \mathbb Z_p \to \mathbb C_p$ if and only if the Mahler coefficients \begin{align*} d_0 &= f(0) \\ d_1 &= f(1)-f(0) \\ d_2 &= f(2)-2f(1)+f(0) \\ &\vdots \\ d_n &= \sum_k (-1)^{n-k} \binom nk f(k) \end{align*} tend to zero in the $p$-adic sense. By finite differences, this is equivalent to the existence of a polynomial $P_e \in \mathbb Q[x]$, taking only integer values on $\mathbb Z$, for which $f(n) \equiv P_e(n) \pmod{p^e}$ holds for $n \ge 0$. (Please correct me if this is wrong!) In either case, we have the explicit formula $$f(x) = \sum_k d_k \binom xk \qquad x \in \mathbb Z_p.$$

In reading proofs of Skolem-Mahler-Lech theorem (e.g. here), I have the following additional questions.

  1. Are there conditions (in terms of either $d_n$ or $P_e$) on when the extended function $f : \mathbb Z_p \to \mathbb C_p$ is analytic?

  2. Assuming $f$ isn't identically zero, are there conditions on which $f$ can have only finitely many zeros? (In particular, the blog post linked seems to claim that analytic is sufficient, but as $\mathbb Z_p$ is quite disconnected I'm confused why this is true.)

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    $\begingroup$ For your last question, analytic power series have finitely many zeroes follows from the Weierstrass preparation theorem. See math.stackexchange.com/a/1417765/68188 $\endgroup$
    – Asvin
    Oct 14 '17 at 16:25
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    $\begingroup$ Analytic in the sense that it has a power series expansion? Or Krasner analytic? $\endgroup$
    – EFinat-S
    Oct 14 '17 at 17:00
  • $\begingroup$ Anyway, I'm pretty sure this appears in Schikhof's or Robert's books. This kind of results are due to Yvette Amice, if I'm correct. If I have time later I'll search. $\endgroup$
    – EFinat-S
    Oct 14 '17 at 17:39
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For your first question: Theorem 54.4 (page 166) in Schikhof's "Ultrametric Calculus" says that $f$ is analytic if and only if

$$\lim_{n\to\infty}\frac{d_n}{n!}=0.$$

Here, analytic means that $f$ has a power series expansion which converges for all $|x|\le1$ (in $\mathbb{Q}_p$, hence in $\mathbb{C}_p$).

Actually, in his book Schikhof gives characterizations in terms of Mahler coefficients of a lot of interesting spaces of $p$-adic functions. Many of them are due to Yvette Amice.

If your meaning of analytic is "Krasner analytic", you may look at Robert's "A course in $p$-adic analysis", section 4.3, page 344.

For your second question, as Asvin points out, a nonzero power series on the closed unit ball has only finitely many zeros there ("Strassman theorem", see Robert, page 306).

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    $\begingroup$ Stress that you mean at the end the closed unit ball $|x| \leq 1$. A nonzero power series converging on the open unit ball $|x| < 1$ could have infinitely many zeros there, such as the $p$-adic logarithm. $\endgroup$
    – KConrad
    Oct 15 '17 at 1:27
  • $\begingroup$ Of course, "unit ball" without open/closed is always confunsing. Thanks. $\endgroup$
    – EFinat-S
    Oct 15 '17 at 1:34

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