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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.

Let $I$ be a complete discrete valuation ring with a continuous ${\mathbb{Z}}_p$-algebra structure. Every continuous function on ${\mathbb{Z}}_p$ with values in $I$ has a unique uniformly convergent expansion in the binomial polynomials, with $a_n={\Delta }^{n}f(0)$. This is the coefficient-ring version of the Mahler theorem. To see that $a_n\to 0$, approximate the function modulo any power of the maximal ideal by a function of period $p^k$. In the expansion of $(E-1{)}^{p^k}$, all intermediate binomial coefficients are divisible by $p$, while $(E^{p^k}-1)$ kills the periodic function. Iterating makes its differences vanish modulo the prescribed ideal power. Finite differences do not increase the uniform norm, so the same conclusion holds for the original function. Binomial inversion gives agreement at the nonnegative integers, and their density in the p-adic integers gives the expansion everywhere. This gives the dual description of p-adic measures by arbitrary bounded sequences of binomial moments, and hence the Amice transform.

The Mahler theorem states that a function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$ is continuous if and only if it has a unique expansionand the expansion converges uniformly. Moreover $\Vert f{\Vert }_{\infty }={\sup }_n|c_n{|}_p$; in particular $f$ takes values in ${\mathbb{Z}}_p$ exactly when all $c_n$ belong to ${\mathbb{Z}}_p$. Here $\left(\genfrac{}{}{0ex}{}{x}{0}\right)=1$ and $\left(\genfrac{}{}{0ex}{}{x}{n}\right)=x(x-1)\cdots (x-n+1)/n!$. These binomial polynomials are continuous and integer-valued on ${\mathbb{Z}}_p$, since they are integer-valued on the dense nonnegative integers. The coefficient condition $c_n\to 0$ therefore makes the series uniformly convergent.

Evaluate at a nonnegative integer $m$. Terms with $n>m$ vanish, giving the finite triangular relation $f(m)={\sum }_{n=0}^m\left(\genfrac{}{}{0ex}{}{m}{n}\right)c_n$. Binomial inversion gives the Mahler coefficientsTo obtain the generating function, multiply the last finite identity by $T^n/n!$ and compare coefficients. Setting $n=j+k$ yieldsThe Mahler coefficient exponential generating function is an identity in ${\mathbb{Q}}_p[[T]]$: each coefficient is a finite sum. It does not assert convergence for every $p$-adic substitution for $T$.