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From: Phil Carmody on 1 Oct 2009 17:06
Rick Decker <rdecker(a)hamilton.edu> writes:
> Phil Carmody wrote:
>> Rick Decker <rdecker(a)hamilton.edu> writes:
>>
>>> Richard L. Peterson wrote:
>>>> I mean is the identity already known, not the
>>>> proof, which is easy.
>>> Your sequence, 1^5 - 3^5 + 5^5 - 7^5 + ...,
>>> is not in the online encyclopedia of integer sequences.
>>>
>>> However, a generalization of your result, for the alternating
>>> sum of powers of odd integers, is indeed known. Search
>>> for "Swiss-knife function". It's quite interesting.
>>
>> If you mean "The Canonical Generating Function or CGF(z) -- a
>> Swiss-knife function", then it looks a load of codswallop. Then
>> again quoting Chaitin right at the start was bound to prejudice me
>> against it. I gave up before
>> I got to anything lucid.
>>
>> Phil
>
> Actually, Phil, I wasn't referring to Huen's work, which I'll
> agree is a bit impenetrable. Instead, I was talking about
> Peter Luschny's 2008 result on what might more properly be
> called Swiss-knife _polynomials_ (though you can still get
> this by having Google search for "Swiss-knife functions";
> it's just a bit farther down the list).
>
> Sorry for any confusion.

*phew*!

Phil
--
Any true emperor never needs to wear clothes. -- Devany on r.a.s.f1
From: Richard L. Peterson on 1 Oct 2009 14:50
Thanks
From: Richard L. Peterson on 8 Oct 2009 19:20
Let Sr(n)=1-3^r+5^r-...+{(-1)^(r+1)}*(2n-1)^r.
Then S1(n)=+-n=+-T1(n) and S3(n)=+-T3(n), where
T1 and T3 are the first and third Tschebycheff polynomials. But S2(n) equals +-T2(n) only for
odd n. For even n it's less by 1.Now S5(n),
which was discussed in this thread, differs from
+-T5(n) by 20n(n^2-1), which is not that much.
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