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:

<math>(x)_k=x(x-1)(x-2)\cdots(x-k+1).</math>

Denote by Δ the forward difference operator defined by

<math>(\Delta f)(x)=f(x+1)-f(x).</math>

Then we have

<math>\Delta(x)_n=n(x)_{n-1}</math>

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:

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

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.fr:Théorème de Mahler