Mahler expansion of the parity function ($(-1{)}^x={\sum }_{n\ge 0}(-2{)}^n\left(\genfrac{}{}{0ex}{}{x}{n}\right)$ on ${\mathbb{Z}}_2$)

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.