# Expansion of the falling factorial

Let $$(x)_n=x(x-1)\ldots (x-(n-1))$$ be the falling factorial.

For small $$n$$ I have found $$(xy)_1=(x)_1 (y)_1,\\ (xy)_2=x (x)_1 (y)_2+(x)_2 (y)_1,\\ (xy)_3=x^2 (x)_1 (y)_3+3 x (x)_2(y)_2+(x)_3(y)_1$$ Is there any general formula for the expansion $$(xy)_n$$ in terms of $$(x)_i, (y)_i$$? I hope must be something likes to $$(xy)_n=\sum_{i=1}^n \alpha_i x^{n-i} (x)_i (y)_{n+1-i},$$ for some sequence $$\alpha_i$$. I think it is well-known result.