Prod[1-(n+1)^3, n=1,...,Inf]
Eric W. Weisstein
eww at wolfram.com
Tue Apr 18 17:18:19 CEST 2006
On Tue, 18 Apr 2006, David W. Cantrell wrote:
>> The product is very similar to
>> http://mathworld.wolfram.com/InfiniteProduct.html eqn (23) and might very
>> well have a similar formula for arbitrary power. Any takers?
>
> Sure. But before I get to that, I see that (23) is not attributed to anyone.
> However, I assume that it was sent to you by Paul Abbott based on my post
> <http://groups.google.com/group/sci.math/msg/de92321b4a19e8e8> on 2006 Mar.
> 29 in the sci.math thread "infinite product". Unless that general result was
> stated previously elsewhere (which it certainly may have been),
(I hadn't seen it before, but if someone knows a reference, I'd be
interested in learning about it.)
> shouldn't it be attributed to me?
Hi, David.
Sorry about that. Yes, it should (and will be in the next update).
> Now to your current question, Eric. (I had already worked out the answer
> before I saw your post here, BTW.)
:)
> Product[1 - 1/n^p, {n, 2, Infinity}]
>
> simplifies, if p is odd, to
>
> 1/(p * Product[Gamma[- (-1)^(j*(1 + 1/p))], {j, 1, p - 1}])
>
> and, if p is even, to the elementary
>
> Product[Sin[Pi*(-1)^((2*j)/p)]/(Pi*I), {j, 1, p/2 - 1}] / p.
>
> I will also be posting this general result to the previously mentioned
> sci.math thread soon.
Very nice. I'll add that additional result (with attribution) to the
InfiniteProduct page for the next update.
Best wishes,
-Eric
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