What does $\displaystyle\prod_{n\geq 1} \frac{n-z}{n+z}$ converge to?

Question

Does the infinite product $$\prod_{n\geq 1} \frac{n-z}{n+z}$$ converge, and if so to what?

It seems that $$\lim_{n\rightarrow\infty}\frac{n-z}{n+z} = 1$$ so it is reasonable to think that the product might converge.

Answer 1

Let $r\ge 2$ and $\zeta_r=e^{2\pi i/r}$. Then quotients of the Gamma function yield the following infinite products, $$ \prod_{n\ge 0}\frac{n^r-z^r}{n^r+z^r}=\prod_{j=1}^{2r}\Gamma(z\cdot \zeta_{2r}^j)^{{(-1)}^{j+1}}, $$ if $z$ is not a nonnegative integer, and $$ \prod_{n\ge 0, n\neq m}\frac{n^r-m^r}{n^r+m^r}=(-1)^m m!\frac{2m}{r}\prod_{j=1}^{2r-1}\Gamma(-m\cdot \zeta_{2r}^j)^{{(-1)}^{j+1}}, $$ if $z=m$ is a nonnegative integer.

Edit: This is not an answer for $r=1$, where we do not have convergence, but nevertheless a nice formula for $r\ge 2$.