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:

$(x)_{k}=x(x-1)(x-2)\cdots (x-k+1).$

Denote by Δ the forward difference operator defined by

$(\Delta f)(x)=f(x+1)-f(x).$

Then we have

$\Delta (x)_{n}=n(x)_{n-1}$

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:

$f(x)=\sum _{k=0}^{\infty }{\frac {(\Delta ^{k}f)(0)}{k!}}(x)_{k}.$

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 specified field, the same identity holds.