For a field \(K\) of characteristic 0 the map \[ XK[[X]] \longrightarrow 1 +XK[[X]], g \mapsto h :=\exp (g) \] defines an isomorphism of groups, additive resp. multiplicative. The central theorem of the article relates, in the case \(K=\mathbb{Q}_p\) with \(p \ge 3\), the coefficient sequence \(f(n)\) of the series \[ h= 1+\sum_{n=1}^\infty f(n)\frac{X^n}{n!} \] to the coefficients \(m_k\) of \[ g=\sum_{k=1}^\infty m_k \frac{X^k}{k} \] as follows: Assume that \(m_k \in \mathbb{Z}_p\) for \(k \ge 2\). Then the function \[ f: \mathbb{N} \longrightarrow \mathbb{Q}_p, n \mapsto f(n), \] is \(p\)-adically continuous, i.e., extends to a continuous function \(\hat f: \mathbb{Z}_p \longrightarrow \mathbb{Q}_p\) if and only if \(|m_1-m_p|_p <1\).
A series of examples is discussed, in particular functions counting the number of suitable permutations, derangemants and arrangemants, in the symmetric group on \(n\) letters. They are related to the values of the “incomplete \(\Gamma\)-function” \(\Gamma (n,r)\) for \(n, r \in \mathbb{N}\). That function admits a \(p\)-adic interpolation \[ \Gamma_p: \mathbb{Z}_p \times 1 + p\mathbb{Z}_p \longrightarrow \mathbb{Z}_p, \] which is described by an explicit formula.