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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.

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