$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed expressions are known to include $\cosh(.)$ and $\pi$.

I hope to use this one day to find a closed expression of $\zeta(3)$. If we could compute $$\prod_{i=1}^{\infty}{1+(\frac{z}{i})^3}$$ for complex z, it follows that $\zeta(3)$ is the coefficient for $z^3$ in its expansion.

Solution 1:

Here is a closed form for the product

$$ {\frac {1}{{z}^{2}\Gamma \left( z+1 \right) \Gamma \left( -\frac{z}{2}-\frac{iz\sqrt {3}}{2} \right) \Gamma \left( -\frac{z}{2}+\frac{iz\sqrt {3}}{2} \right)}},\quad i=\sqrt{-1},$$

where $\Gamma(z)$ is the gamma function.

Solution 2:

Just as an addendum to Robert Israel's comment. The Taylor expansion of the formula given in Mhenni Benghorbal's answer is $$1+\zeta (3) z^3 + \left(\frac{\zeta (3)^2}{2}-\frac{\pi ^6}{1890}\right)z^6+(\frac{1}{6} \left(\zeta (3)^3+2 \zeta (9)\right)-\frac{\pi ^6 \zeta (3)}{1890})z^9 +O\left(z^{12}\right)$$ which I find quite nice.