Let $X$ and $Y$ be two independent random variables with Poisson distribution, $F_2(y,\mu_2)$ be the Poisson CDF of $Y$ with mean $\mu_2$ and $P_1(x,\mu_1)$ be the Poisson PMF of $X$ with mean $\mu_1$. How to rewrite the following expression in terms of $F_1$,$F_2$,$P_1$,$P_2$, $\mu_1$, $\mu_2$ and ... in order to get rid of sigma ($\Sigma$)?
\begin{equation} \sum_{t=\alpha-\beta+1}^{\alpha} F_2(\alpha-t,\mu_2)P_1(t,\mu_1) \end{equation}
\begin{align*} \sum_{t=\alpha-\beta+1}^{\alpha} F_2(\alpha-t,\mu_2)P_1(t,\mu_1) &=\sum_{t=\alpha-\beta+1}^{\alpha}P(Y \le \alpha -t) P(X =t)\\ &=\sum_{t=\alpha-\beta+1}^{\alpha}P(Y \le \alpha -t,\,X =t)\\ &=\sum_{t=\alpha-\beta+1}^{\alpha}P(X=t,\, X+Y \le \alpha)\\ &=P(\alpha-\beta+1\le X\le \alpha,\, X+Y \le \alpha). \end{align*}