According to a result proven in the last part, any factorial polynomial of the $n$th degree can be expressed as a usual polynomial of the $n$th degree and vice versa.
The coefficients occurring when stating this result for the simplest factorial polynomial $x^\underline{n}$ and the simplest polynomial $x^n$ of the $n$th degree. Because of their interesting properties, those coefficients deserve the following separate definition.
Let $n\ge 1$ and $x^\underline{n}$ be a factorial polynomial of the $n$th degree and let $x^n$ be a polynomial of the $n$th degree.
The coefficients $\left[\begin{array}{c}n\\r\end{array}\right]$ and $\left\{\begin{array}{c}n\\r\end{array}\right\}$ occurring in the linear combinations. $$x^\underline{n}= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right](-1)^{n-r}x^r$$ and $$x^{n}= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}x^\underline{r}$$
are uniquely determined when expressing factorial polynomials via polynomials and vice versa. $\left[\begin{array}{c}n\\r\end{array}\right]$ are called Stirling numbers of the first kind and $\left\{\begin{array}{c}n\\r\end{array}\right\}$ are called Stirling numbers of the second kind.
Chapters: 1 2
Explanations: 3 4
Lemmas: 5
Proofs: 6 7 8 9
Propositions: 10 11 12 13