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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.

Definition: Stirling Numbers of First and Second Kind

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.

| | | | | created: 2020-04-04 10:25:50 | modified: 2020-04-05 18:49:24 | by: bookofproofs | references: [8404]

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Bibliography (further reading)

[8404] Miller, Kenneth S.: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 1960