How can i make the following change to this infinite series? $$ e^z  - 1  = \sum_{n=1}^\infty \frac {z^n}{n!} $$
Given the above function and its corresponding series expansion, is there anything i could do to the left side of the equation so that the infinite series looks like this instead???
$$ \sum_{n=1}^\infty (\frac {z^n}{n!})^{a} $$
that is to the power of $\mathbf A$ which would be any constant. 
$$ (e^z-1)^a=  \sum_{n=1}^\infty (\frac {z^n}{n!})^{a} $$   
Would it be just like this? ^ 
Thank you very much for your time and help. 

If you put a series to a certain power, it doesn't mean it's equal to the series of each term to that power.
I don't think there is a general expression for this series except when $a$ is 0 or 1.
