One has the nice identities $${xy\choose 1}={x\choose 1}{y\choose 1},$$ $${xy+1\choose 2}={x+1\choose 2}{y+1\choose 2}+{x\choose 2}{y\choose 2}$$ and $${xy+2\choose 3}={x+2\choose 3}{y+2\choose 3}+4{x+1\choose 3}{y+1\choose 3}+{x\choose 3}{y\choose 3}.$$

(The proof is essentially trivial by interpreting ${z\choose k}$ as a polynomial of degree $k$.)

This sequence of identities stops: There seems to be no nice expression of ${xy+k-1\choose k}$ as a linear combination of ${x+k-i\choose k}{y+k-i\choose k},i=1,\ldots,k$ for $k\geq 4$.

*Is there a good reason for this breakdown?* (Probably a better question is: Is there a reason for these identities to hold for $k=2$ and, especially, for $k=3$?)