Trying to Express A Factorial As A Polynomial

Question

I'd like to express the following as a polynomial. $$(a-1)(a-2)(a-3) . . . (a-b)$$ where $b<a$

I'm currently working on it now, but wanted to see if anyone's already done it, or already know what the answer is.

Answer 1

Here are some asymptotic results of mine that might be of some use (although the product goes in the wrong direction):

$$ \lim_{n \to \infty}\dfrac{(x+(n-1)/2)^n-x(x+1)...(x+n-1)}{(x+(n-1)/2)^{n-2}(n^3-n)/24} =1 $$

and

$$ \lim_{n \to \infty} (x+(n-1)/2)\dfrac{(x+(n-1)/2)-(x(x+1)...(x+n-1))^{1/n}} {(n^2-1)/24} =1 $$

An outline of the proofs is here:

Limit of $\sqrt[n]{(x+1)...(x+n)} - x$ as $x \to +\infty$