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

Then we have

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 says that if f is a continuous p-adic-valued function of a p-adic variable, then the analogy goes further:

That as weak an assumption as continuity is enough is remarkable.

It is a fact of algebra that if f is a polynomial function with coefficients in any specified field, the same identity holds.