Mahler coefficient exponential generating function (${\sum }_{n\ge 0}{c}_n{T}^n/n!=e^{-T}{\sum }_{n\ge 0}f(n)T^n/n!$)

For the Mahler coefficients of a continuous function on the p-adic integers, finite binomial inversion gives $c_n={\sum }_{j=0}^n(-1{)}^{n-j}(\genfrac{}{}{0ex}{}{n}{j})f(j)$. Equivalently their exponential generating function is the displayed identity in the formal power-series ring ${\mathbb{Q}}_p[[T]]$. Each coefficient involves only finitely many terms; no claim of convergence for every $p$-adic value of $T$ is needed.