Moment generating function of $Z=X-Y$ where $X,Y$ are independent binomial RVs

Question

$X$ and $Y$ are statistically independent random variables.

• $X\sim \text{Bin}(n_1,p_1)$
• $Y\sim \text{Bin}(n_2,p_2)$
• $Z = X -Y$
• $q_1 = 1 - p_1$
• $q_2 = 1 - p_2$

I am trying to find the moment-generating function $ \phi_z(\omega)$. At this point I got to $(q_1 + p_{1}\cdot e^{\omega})^{n_{1}} \cdot (q_2 + p_2\cdot e^{-\omega})^{n_{2}}$

Is this the correct moment generating function? Attached are my notes about how I got to this result. Notes