Lemma 10.58.5. Suppose that $f: n \mapsto f(n) \in A$ is defined for all $n$ sufficiently large and suppose that $n \mapsto f(n) - f(n-1)$ is a numerical polynomial. Then $f$ is a numerical polynomial.
Proof. Let $f(n) - f(n-1) = \sum \nolimits _{i = 0}^ r \binom {n}{i} a_ i$ for all $n \gg 0$. Set $g(n) = f(n) - \sum \nolimits _{i = 0}^ r \binom {n + 1}{i + 1} a_ i$. Then $g(n) - g(n-1) = 0$ for all $n \gg 0$. Hence $g$ is eventually constant, say equal to $a_{-1}$. We leave it to the reader to show that $a_{-1} + \sum \nolimits _{i = 0}^ r \binom {n + 1}{i + 1} a_ i$ has the required shape (see remark above the lemma). $\square$
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