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.