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Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume that the series $\sum_{n\ge0}a_n+b_n$ converges in $\mathbb Q_p$. Can the function $f$ be continued in a larger disk in an analytic function?

Thanks in advance for for any hint or answer.

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  • $\begingroup$ You are not assuming that $\sum a_nz^n$ and/or $\sum b_nz^n$ converge, right? $\endgroup$ – EFinat-S Mar 9 at 4:20
  • $\begingroup$ $\sum_{n\ge}a_nz^n$ and $\sum_{n\ge}b_nz^n$ converge for $|z|_p<1$ but not $\sum_{n\ge}a_n$ and $\sum_{n\ge0}b_n$. $\endgroup$ – joaopa Mar 9 at 4:33
  • $\begingroup$ Ok, I understand your question now. I deleted my anwer. $\endgroup$ – EFinat-S Mar 9 at 6:06
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I think the answer is "not necessarily", by the following (counter)example. First, let $b_n=-a_n$ for $n\ge0$. Then $\sum(a_n+b_n)=0$ and $$f(z)=\sum_{n\ge0} a_n(1-z)z^n=\sum_{n\ge0} (a_n-a_{n-1})z^n,$$ where we set $a_{-1}=0$. Now, define $$a_n:=\frac{-1}{n+1}.$$ Then, $\sum a_nz^n$ converges if and only if $|z|_p<1$. Also, $$f(z)=\sum_{n\ge0}\frac{1}{n(n+1)}z^n,$$ which converges if and only if $|z|_p<1$.

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