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Mahler's theorem
In the notation of combinatorialists, 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:
It is remarkable that as weak an assumption as continuity is enough.
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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