$\sum\limits_i \binom{i}{a}x^{i-a}\binom{n-i}{b}y^{n-i-b}$

\[\sum_i \binom{i}{a}x^{i-a}\binom{n-i}{b}y^{n-i-b} \\ = [z^n]{z^a\over (1-xz)^{a+1}}{z^b\over (1-yz)^{b+1}} \\ =[z^nu^av^b]\sum_{a,b}{(uz)^a\over (1-xz)^{a+1}}{(vz)^b\over (1-yz)^{b+1}} \\ =[z^nu^av^b]{1\over 1-xz}{1\over 1-yz}{1\over 1-{uz\over 1-xz}}{1\over 1-{vz\over 1-yz}} \\ =[z^nu^av^b]{1\over 1-xz-uz}{1\over 1-yz-vz} \\ =[z^nu^av^b]{1\over 1-(x+u)z}{1\over 1-(y+v)z} \\ =[z^nu^av^b]({1\over 1-(x+u)z}-{1\over 1-(y+v)z}){1\over (x+u)z-(y+v)z} \\ =[z^{n+1}u^av^b]({1\over 1-(x+u)z}-{1\over 1-(y+v)z}){1\over (x+u)-(y+v)} \\ =[u^av^b]((x+u)^{n+1}-(y+v)^{n+1}){1\over (x-y)+(u-v)} \]