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

[Dean Hickerson]

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?

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.

Dean Hickerson
dean at math.ucdavis.edu