Amice transform (${\mathcal{A}}_{\mu }(T)={\int }_{{\mathbb{Z}}_p}(1+T{)}^{x}d\mu (x)$)

The coefficient of $T^n$ is $\int \left(\genfrac{}{}{0ex}{}{x}{n}\right)d\mu (x)$. The Mahler expansion of continuous p-adic functions identifies $I$-valued p-adic measures on ${\mathbb{Z}}_p$ with $I[[T]]$. If $D=(1+T)d/dT$, then $D^k{\mathcal{A}}_{\mu }(0)=\int x^{k}d\mu (x)$. A measure supported on ${\mathbb{Z}}_p^{\times }$ has ${\sum }_{{\zeta }^p=1}{\mathcal{A}}_{\mu }(\zeta (1+T)-1)=0$, because the sum of ${\zeta }^x$ is zero for $p\nmid x$.