The main question I want to ask is inspired from this question
Find the value of $$\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)$$
Now, I have solved this question easily using product representation of $\displaystyle \frac {\sin x}{x}$ and then substituting $x=i\pi$ in that representation to get $$\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)=\frac {\sinh \pi}{\pi}$$
But this led me to investigate a more general infinite product systems as $$\displaystyle\xi_p= \prod_{n=1}^{\infty} \left(1+\frac {1}{n^p}\right)$$ where $p\in N$ and $p\ge 2$. But I don't know of any special function like that of $\displaystyle \frac {\sin x}{x}$ that could be used to calculate $\xi_p$, so I just started with finding values of $\xi_2,\xi_3,\xi_4,\cdots$, and what is quite surprising is that I could conjecture beautiful closed forms for odd and even $p$ (which I still think are the same). Here are the closed forms I could conjecture
$\star$For odd $p$ $$\displaystyle\xi_p=\frac {1}{\displaystyle \prod_{r=1}^{s-1} \Gamma\left(1+(-1)^r (-1)^{\frac rs}\right)}$$
$ \star$While for even $p=2k$,$k\ge 1$ and $k\in N$ I got $$\displaystyle \xi_p=\alpha \frac {\displaystyle \prod_{r=1}^k \sin\left((-1)^{\frac {2r-1}{s}}\pi\right)}{\pi^k}$$ where $\alpha$ can take only the values $+1,-1,i,-i$ depending on some condition (which I was not able to find)
It did irk me that the closed form for even $p$ had product of sines instead of gamma function and noticed that using the properties of gamma function I can convert the RHS of the case of even $p$ to a similar form of RHS of odd $p$ but got stuck between at some point.
I would like to know whether or not my closed forms are correct and also some rigorous methods to find the closed forms for $\xi_p$.
I would also appreciate if someone could give a proof for the closed forms that he/she would have got for $\xi_p$