Series including binomial coefficients I am trying to solve the following equation:
$\Sigma_{n=0}^N\binom{N}{n} e^{cn( x + y)}$
I know that you can write $\Sigma_{n=0}^N\binom{N}{n}=2^n$ but I cannot do this right now because I have the exponential factor.
Then I though of expressing $ e^{cn( x + y)}= \Sigma_{n=0}^N \frac{[cn(x + y)]^n}{n!} $
But this only complicates things further.
Any tips on how I should proceed ?
Edit: c,y,x are constants

Substitute $z=e^{c(x+y)}$
$\sum_{k=0}^{N} \binom{N}{k}e^{ck(x+y)}=\sum_{k=0}^{N} \binom{N}{k}z^k={(1+z)}^N={(1+e^{c(x+y)})}^N$
$\sum_{k=0}^{N} \binom{N}{k}z^k={(1+z)}^N$
The above statement is just from Newton's binomial theorem if we plug $x=1$ and $y=z$:
https://en.wikipedia.org/wiki/Binomial_theorem
I would say that this is a pretty nice form
