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.
Referenced By
Difference operator | List of mathematical topics (M-O) | List of number theory topics | P-adic analysis
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