# 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

*n*th term is

*x*^{n}.

**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.