Editing Mahler's theorem (section)

== Statement ==
Let <math>(\Delta f)(x)=f(x+1)-f(x)</math> be the forward [[difference operator]]. Then for any ''p''-adic function <math>f: \mathbb{Z}_p \to \mathbb{Q}_p</math>, Mahler's theorem states that <math>f</math> is continuous if and only if its [[Newton series]] converges everywhere to <math>f</math>, so that for all <math>x \in \mathbb{Z}_p</math> we have

:<math>f(x)=\sum_{n=0}^\infty (\Delta^n f)(0){x \choose n},</math>

where

:<math>{x \choose n}=\frac{x(x-1)(x-2)\cdots(x-n+1)}{n!}</math>

is the <math>n</math>th [[binomial coefficient]] polynomial. Here, the <math>n</math>th forward difference is computed by the [[binomial transform]], so that<math display="block"> (\Delta^n f)(0) = \sum^n_{k=0} (-1)^{n-k} \binom{n}{k} f(k).</math>Moreover, we have that <math>f</math> is continuous if and only if the coefficients <math>(\Delta^n f)(0) \to 0</math> in <math>\mathbb{Q}_p</math> as <math>n \to \infty</math>.

It is remarkable that as weak an assumption as continuity is enough in the ''p''-adic setting to establish convergence of Newton series. By contrast, Newton series on the field of [[complex numbers]] are far more tightly constrained, and require [[Carlson's theorem]] to hold.