Welcome guest
You're not logged in.
271 users online, thereof 1 logged in

The following lemma shows that formulas involving Stirling numbers of the first and second kind and factorial powers can be easily transformed into each other.

Lemma: Stirling Numbers and Rising Factorial Powers

From the definition of Stirling numbers of the first and second kind that uses the falling factorial powers it follows for the rising factorial powers

$$x^\overline{n}= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right]x^r$$


$$x^{n}= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^{n-k}x^\overline{r}$$

for all natural numbers $n\ge 1.$

| | | | | created: 2020-04-05 18:37:43 | modified: 2020-04-05 19:06:32 | by: bookofproofs | references: [1112]

1.Proof: (related to "Stirling Numbers and Rising Factorial Powers")

Edit or AddNotationAxiomatic Method

This work was contributed under CC BY-SA 4.0 by:

This work is a derivative of:

Bibliography (further reading)

[1112] Graham L. Ronald, Knuth E. Donald, Patashnik Oren: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition