We know that the following Pochhammer symbol $$(-a-x-1)_n=(-a-x-1)(-a-x)(-a-x+1)...(-a-x-1+n-1)$$ is a polynomial of degree $n$, that is,
$$(-a-x-1)_n=\sum_{k=0}^na_k\,x^k$$
I wonder if it is possible to get a closed formula for the coefficients $ a_k $.
It's clear that $a_n=(-1)^n$ but I don't know how to obtain the others.
Any help will be welcomed.
Hint: In terms of the Stirling numbers of the first kind, $$ ( - a - x - 1)_n = \sum\limits_{k = 0}^n ( - 1)^n s(n,k)(a + x+1)^k . $$ Now expand $(a+1+x)$ in powers of $x$ using the binomial theorem and re-arrange the sum to obtain a formula for the $a_k$'s.