Mahler's theorem

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In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses continuous p-adic functions in terms of polynomials.

In any field, one has the following result. Let

(\Delta f)(x)=f(x+1)-f(x)\,

be the forward difference operator. Then for polynomial functions f we have the Newton series:

f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},

where

{x \choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}

is the kth binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity.

Mahler's theorem states that if f is a continuous p-adic-valued function on the p-adic integers then the same identity holds.

The relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose kth term is xk.

It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the complex number field are far more tightly constrained, and require Carlson's theorem to hold.

It is a fact of algebra that if f is a polynomial function with coefficients in any field of characteristic 0, the same identity holds where the sum has finitely many terms.

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