Mahler's theorem
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In the notation of combinatorialists, which conflicts with that used in the theory of special functions, the Pochhammer symbol denotes the falling factorial:
- <math>(x)_k=x(x-1)(x-2)\cdots(x-k+1).<math>
Denote by Δ the forward difference operator defined by
- <math>(\Delta f)(x)=f(x+1)-f(x).<math>
Then we have
- <math>\Delta(x)_n=n(x)_{n-1}<math>
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:
- <math>f(x)=\sum_{k=0}^\infty\frac{(\Delta^k f)(0)}{k!}(x)_k.<math>
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 field of characteristic 0, the same identity holds.fr:Théorème de Mahler