Help simplifying a product

Question

I have this "product of products": $$\prod_{k=1}^{x}\prod_{i=1}^{k}(x-f(i))$$ Is there any way to simplify it respect to $x$? Thank you!

Answer 1

The factor $x-f(i)$ appears once for every $k\geq i$, so it equals

$$(x-f(1))^{x}(x-f(2))^{x-1}\cdots(x-f(x-1))^2\cdot(x-f(x))^1.$$

There is no way to simplify it further if $f$ is unknown. One could expand it and find explicit formulas for all terms, but that won't give a simpler expression. Compare with $a^3+3a^2b+3ab^2+b^3$ vs. $(a+b)^3$.