Mahler's theorem
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In the notation of combinatorists, which conflicts with that used in the theory of special functions, the Pochhammer symbol denotes the falling factorial:
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:
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.
