Can anyone help me out trying to simplify the left hand side of the below equation to obtain the right hand side?
$$ \displaystyle\prod_{\substack{n=-\infty \\n\neq 0}}^\infty\left(1-\frac{z}{n}\right)e^{z/n} = \prod_{n=1}^\infty\left(1-\frac{z^2}{n^2}\right). $$
Note that $$\left(1-\frac z n\right) e^{z/n} \cdot \left(1+\frac z n\right) e^{-z/n} = 1-\frac{z^2}{n^2}.$$