Given an entire function $\prod_{n=0}^∞ E_n (\frac{z}{z_n})$ , show that $(z_n )_n∈N$ is a complete list of the zeroes of this function in which each zero appears as many times as its multiplicity.
Thanks in advance!
Edit: Sorry, I neglected to mention that $E_n(z) = (1 − z)e^{z+\frac{z^2}{2}+...+\frac{z^n}{n}}$ where n is a positive integer.