Sum of product of combinations

Question

I am trying to find a direct way (i.e. without explicitly doing the sum) of computing:

$$\sum_{i=0}^{min(a,\,b)}{a \choose i}{b \choose i}$$

for a given $a$, $b$.

Is there a closed form way of computing this sum? Can somebody help me?

Answer 1

Expanding on my comment, if $a\le b$ then $$\sum_{i=0}^a\binom{a}{i}\binom{b}{i} =\sum_{i=0}^a\binom{a}{a-i}\binom{b}{i} =\binom{a+b}a$$ by the Vandermonde identity.