Mahler coefficient ($a_n=({\Delta }^{n}f)(0)$)

For a continuous function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$, its nth Mahler coefficient isThis is the explicit iterate of the forward difference operator. These coefficients are exactly those of its Mahler expansion.

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.

The locally constant function equal to one on $2{\mathbb{Z}}_2$ and minus one on its other coset has Mahler coefficients $a_n=(-2{)}^n$: at nonnegative integers its values are $(-1{)}^j$, so the forward difference operator multiplies it by $-2$. Since $|a_n{|}_2=2^{-n}$ and the binomial polynomials have absolute value at most one on the 2-adic integers, the series converges uniformly. The binomial theorem verifies its values on the nonnegative integers, and continuity extends the identity to the entire domain.

Approximate a continuous function on ${\mathbb{Z}}_p$ uniformly by a function constant on residue classes modulo $p^r$. On this finite space, translation $S$ satisfies $S^{p^r}=I$, so ${\Delta }^{p^r}=(S-I{)}^{p^r}$ has every matrix coefficient divisible by $p$. Hence $\Vert {\Delta }^{j{p}^r}\Vert \le p^{-j}$, and the Mahler coefficients of a locally constant function tend to zero. Since $|a_n(f)-a_n(g){|}_p\le \Vert f-g{\Vert }_{\infty }$, uniform approximation proves the assertion for every continuous function. This completes the coefficient-decay part of the Mahler theorem.