I'm trying to to make the below expression simpler, and it would be great if it could be expressed as something like $(x+y)^k$.
$$ \sum_{i=0}^k\binom{n+1}i\binom{m+1}{k-i}x^iy^{k-i} $$
The number $\binom{n+1}i$ is the coefficient of $x^i$ in $(1+x)^{n+1}$, and similarly $\binom{m+1}{k-i}$ is the coefficient of $y^{k-i}$ in $(1+y)^{m+1}$. The expression above is in fact the $k$th Chern class $c_k$ of the direct sum of two projective spaces ($CP^n$ and $CP^m$), but the problem may be viewed as completely separate from differential geometry.
I've tried to see if the coefficients match up with coefficients in expansions like $(1+x+y)^{n+m}$, but I can't see any immediate pattern. So, my question is, is it possible to simply the above expression?
This is the coefficient of $t^k$ in the polynomial $(1+tx)^{n+1}\cdot(1+ty)^{m+1}$.
Not sure one can go further in full generality.