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.



Analytic continuation of a $p$-adic function
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 $|z|\_p<1$.


