Can $\sum_{i=a}^{b}p^{i}\binom{i}{a}\binom{b}{i}$ be simplified? I'm trying to compute a complicated thing, and I end up with terms like:
$$\sum_{i=a}^{b}p^{i}\binom{i}{a}\binom{b}{i}$$
$a$ and $b$ are nonnegative integers, $0<p<1$.
I don't see a way to make this simpler. Am I missing something or is this the simplest thing that I'm going to get?

Yes, it can be simplified. Note that for $0\leq a\leq b$,
$$\binom{i}{a}\binom{b}{i}=\binom{b}{a}\binom{b-a}{i-a}.$$
Hence
$$\sum_{i=a}^{b}p^{i}\binom{i}{a}\binom{b}{i}=p^a\binom{b}{a}\sum_{i=a}^{b}\binom{b-a}{i-a}p^{i-a}.$$
Can you take it from here?
