Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-Leffler"? By that I mean that $f(x)$ can be expressed as $f(x)=g(x)+\sum_{n=0}^{\infty} \frac{a_n(x)}{h(x)^n}$, where $g(x)$ converges in a disk greater than the unit ball, $h(x)$ is a polynomial of norm $\le 1$, and the $a_n(x)$ are polynomials satisfying $\text{ord}_p(a_n)\ge rn-m$ for some $r>0$ and a constant $m \in \mathbb{R}$. Here the order of a polynomial is the minimum order of its coefficients.

I am particuarly interested in the values $f(\lambda)$, for Teichmuller representatives $\lambda$, which satisfy $\lambda^q=\lambda$ for some $q=p^r$. Thus I would like $h(x)$ not to have a zero at the Teichmuller representatives. Apparently the values at the Teichmuller representatives are related to classical results on Gauss Sums but I couldn't find anything in the literature.

  • $\begingroup$ By "analytic continuation", you mean that the power series converges also outside the unit ball, or analytic in the sense of Krasner? $\endgroup$
    – EFinat-S
    Feb 13 '20 at 15:09
  • $\begingroup$ @EFinat-S, I have clarified what I mean by analytic continuation, I think it corresponds to the latter. $\endgroup$ Feb 13 '20 at 16:11
  • $\begingroup$ Ok, that makes an important difference. I'll delete my answer. $\endgroup$
    – EFinat-S
    Feb 13 '20 at 16:16
  • $\begingroup$ Sorry for another question: is the condition $\operatorname{ord}_p(a_n) \ge r n$ a condition on some coefficient of $a_n(x)$, all coefficients of $a_n(x)$, some value of $a_n(x)$, or all coefficients of $a_n(x)$, or is it something else? $\endgroup$
    – LSpice
    Feb 13 '20 at 18:54
  • $\begingroup$ I don't know if this will help you, but the book "Analytic elements in $p$-adic analysis" by Alain Escassut has a chapter called "Composition of anaytic elements". Perhaps this is useful since you are composing an analytic function with a rational function. $\endgroup$
    – EFinat-S
    Feb 13 '20 at 23:38

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