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From: Rob Johnson on 16 Jul 2010 04:07
In article <l2czkxspdct.fsf(a)shaggy.csail.mit.edu>,
Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote:
>Rob Johnson <rob(a)trash.whim.org> wrote:
>>
>> [...] The Theorem and Lemma above not only provide a proof that if
>> a polynomial of degree n takes n+1 consecutive integers to integers,
>> then it can be written as an integral linear combination of
>> combinatorial polynomials, and therefore, takes all integers to
>> integers, but also gives a simple formula for the coefficients of
>> that integral linear combination.
>
>That's many centuries old and very well-known, cf. the reference
>to "finite difference calculus" in my earlier reply. For a much
>more general modern treatment google "umbral calculus".

I looked up a few sites regarding umbral calculus. It looks
interesting, but I may need to look at a few more and let it
sink in. Thanks.

By the reference to "finite difference calculus" in your earlier
reply, did you mean the mention of finite difference calculus in
<http://groups.google.com/group/sci.math/msg/53e946bb51be0399>, or
were you intending something more specific? The article referenced
<http://deltaepsilons.wordpress.com/2009/07/21/integer-valued-polynomials/>
seems to cover similar material to my post, but stops before getting
to the Umbral Taylor Series.

Actually, the corollary I posted is what the Wolfram site calls the
Umbral Taylor Series. What the Wikipedia site calls the Umbral
Taylor Series is closer to the Maclaurin Series.

Rob Johnson <rob(a)trash.whim.org>
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