Prod[1-(n+1)^3, n=1,...,Inf]

[Eric W. Weisstein]

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