In the notation of combinatorists, which conflicts with that used in the theory of special functions, the Pochhammer symbol denotes the falling factorial:

$(x)\_k=x(x-1)(x-2)\backslash cdots(x-k+1).$

Denote by Δ the forward difference operator defined by

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

Then we have

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

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; the Newton series holds:

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

It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the complex number field are far more tightly constrained, and require Carlson's theorem to hold.

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.

Factorial and binomial topics | Mathematical theorems

Théorème de Mahler