Mahler's theorem

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In the notation of combinatorialists, which conflicts with that used in the theory of special functions, the Pochhammer symbol denotes the falling factorial:

(x)_k=x(x-1)(x-2)\cdots(x-k+1).

Denote by Δ the forward difference operator defined by

(Δf)(x) = f(x + 1) − f(x).

Then we have

Δ(x)n = n(x)n − 1

so that the relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose nth term is xn.

Mahler's theorem, named after Kurt Mahler (1903–1988), says that if f is a continuous p-adic-valued function of a p-adic variable, then the analogy goes further; the Newton series holds:

f(x)=\sum_{k=0}^\infty\frac{(\Delta^k f)(0)}{k!}(x)_k.

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.

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