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I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.
Perhaps to a working analyst that argument looks natural, but to me it is extraordinary that anybody could come up with that!
Possible typo: should “$H_n x = 0$” be “$\phi^{(k-i)}(H_n x)=0$”?
I have changed
[[falling power]]
to
[[falling factorial]]
which seems to be what was intended.
I myself usually call it a “falling power”, and I believe this is the term used by Graham, Knuth, and Patashnik, so I added that terminology to falling factorial as well as a few redirects.
I'm glad that we do have a page on the falling power after all. (To my mind, it's not so much a falling factorial as a partial factorial. Factorials are always falling, after all.)
Mike, thanks for catching the typo.
I wasn’t aware of this result before; it surprised me somewhat!
This is a special case of a theorem that says when the restriction map $C^\infty(M) \to C^\infty(K)$, for $M$ a manifold and appropriate closed subspaces $K\subset M$, is a submersion. For $M=\mathbb{R}^n$ one has the improved result that it is a split surjection of Fréchet spaces.
The theorem at hand considers $K=\{0\}\subset \mathbb{R}=M$, and by smooth functions on a point one has to consider Whitney jets, which reduce to a mere sequence of numbers, but in general need to satisfy the kind of error estimates that Taylor’s theorem gives.
David, might you have a spare minute to add this kind of comment to the entry?
Sure. I might be writing a short note about a small extension I have made (from Euclidean space, which is in the literature, to Riemannian manifolds of bounded geometry) at some point, so I should gather the relevant references etc before then :-)
I would have called it a shifted factorial, since the difference between it and $k!$ is that it starts at $x$ rather than at $k$.
Mike, yes, that also makes sense, especially when $x$ is not a natural number.
Of course, there's also the case when $k$ is negative; then it's a rising power and a shifted reciprocal factorial. (In particular, $1/k! = 0^{\underline{-k}}$.)
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