Let $p$ be a prime. While I am proving the convergence of a series expansion defined by $$ \mathfrak{F}(x)=\exp(\pi (x-x^p))=\sum_{n=0}^\infty c_n (\pi x)^n $$ over $\mathbb{C}_p$, where $\pi^{p-1}=-p$ and $c_n$ is defined by the recurrence relation $$ c_0=1, c_{-p}=\cdots=c_{-1}=0, nc_n=c_{n-1}+c_{n-p}, $$ I found that the convergence of the series on the disk of radius $1+\epsilon>1$ is equivalent to the fact that the p-adic valutation of the sequence $c_n$ is smaller than or equal to 1. I tried some evaluation for $c_n$ when $p=7$ but the numerator gets uncontrollable as $n$ gets larger. It must be enough to prove for the case when $p$ divides $n$, but even when $p=n$, this claim seems nontrivial since $p$ divides $(p-1)!+1$, which is obviously true by some theorems from group and field theory, but the proof seems quite nontrivial and impossible to applicate for the general case.
Is my approach even correct? I doubt if the proof must be the opposite way: proving the convergence of the series implies that the p-adic valuation for the sequence is smaller than or equal to 1. Please give me any hint.
Thanks in advance.