= Mahler expansion of the parity function
{c}
{title2=$(-1)^x=\sum_{n\geq0}(-2)^n\binom xn$ 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.
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