Two experiments with different probabilities of success run independently $n_1$ and $n_2$ times. I'm modeling the number of successes of each experiment as two independent binomial random variables: $X\sim\mathcal B(n_1, p_1)$ and $Y\sim\mathcal B(n_2, p_2)$.
I would like to know $\Pr[X + Y = k]$ for a constant $k$, i.e., the probability that the sum of the successes of the two experiments in $n_1+n_2$ trials is $k$.
Is there an expression in terms of $n_i$, $p_i$, and $k$ for such probability?
It will be the convolution: $$\mathsf P(X+Y=k)=\sum_{r=0}^k \mathsf P(X=r)~\mathsf P(Y=k-r)\\=\sum_{r=0}^k \binom{n_1}r\binom{n_2}{k-r}{p_1}^r{p_2}^{k-r}(1-p_1)^{n_1-r}(1-p_2)^{n_2-k+r}$$
Which will not close neatly, unless $p_1=p_2$.