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

[Eric W. Weisstein]

On Mon, 17 Apr 2006, Dean Hickerson wrote:

> Zak Seidov wrote:
>
>> Prod[1-(n+1)^3,n=1,...,Inf]=
>> Cosh[(Sqrt[3]*Pi)/2]/(3*Pi)=
>> 0.809396597366290109578680478726.
>
> The exponent should be -3, not 3.  (And it could be simplified to
> prod[1-n^-3, n=2,...,inf].)
>
>> Is this cons known?

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?

Cheers,
-E

> Not to me.  But Mathematica is able to evaluate the infinite product in
> closed form in less than a second, so presumably it's not hard to prove.
> Mma also gives closed forms for the product with the exponent replaced
> by other negative integers, in terms of trig, hyperbolic trig, and gamma
> functions.