I know that:

$\displaystyle

\sum _{i=0}^{\infty } \frac{ X^i }{i!}e^{-X} \sum _{m=0}^{N-1+i} \frac{Y^m}{m!}e^{-Y}

$

and

$\displaystyle \sum _{m=0}^{N-1} \frac{Y^m}{m!}e^{-Y}+\sum _{m=N}^{\infty } \frac{Y^m}{m!}e^{-Y}\left(1- \sum _{i=0}^{m-N} \frac{ X^i }{i!}e^{-X}\right)$

should be equivalent, but how do you get from the first form to the second? Y and X are positive and real.

Cheers