Mathematica answers

    Sum[Binomial[a, k - i]*Binomial[b + i, i], {i, 0, k},Assumptions -> a >= k | b > 0]
$$\binom{a}{k} \, _2F_1(b+1,-k;a-k+1;-1) $$
Addition. Maple performs

    sum(binomial(a, k-i)*binomial(b+i, i), i = 0 .. k)assuming a>=k,b>0
$${a\choose k}{\mbox{$_2$F$_1$}(-k,b+1;\,a-k+1;\,-1)} $$
I think both answers are equivalent.