Falling Factorial

Defintion

For $k\ge 1$ we define $(x{)}_k:=x(x-1)...(x-k+1)$ called Falling Factorial for $x$.

$(n{)}_k=x!(\genfrac{}{}{0ex}{}{n}{k})\frac{n!}{(n-k)!}$

Vector space of polymonids of drgree $\le d$, Dimension $=d+1$

$$\begin{array}{rl}(x{)}_0& =1\\ (x{)}_1& =x\\ (x{)}_2& =x^2-x\\ (x{)}_3& =x^3-3x^2+2x\end{array}$$

Monomial basis: $(1,x,...,x^d)$
Falling factorial also form a basis for this space

Stirling Matrix