Proof
(related to Lemma: Stirling Numbers and Rising Factorial Powers)
- Let $n\ge 1$ be a natural number.
- By definition of the Stirling numbers of the first kind, that involves falling factorial powers $$x^\underline{n}= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right](-1)^{n-r}x^r.$$
- By virtue of the reciprocity law for factorial powers, we can transform this into a formula involving rising factorial powers as follows $$\begin{align}
(-1)^n(-x)^\overline{n}&= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right](-1)^{n-r}x^r\quad\Longleftrightarrow\nonumber\\
(-1)^nx^\overline{n}&= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right](-1)^{n-r}(-x)^r\quad\Longleftrightarrow\nonumber\\
(-1)^nx^\overline{n}&= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right](-1)^{n-r}(-1)^rx^r\quad\Longleftrightarrow\nonumber\\
(-1)^nx^\overline{n}&= (-1)^{n}\sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right]x^r\nonumber\\
x^\overline{n}&= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right]x^r.\nonumber\\
\end{align}$$
- By definition of the Stirling numbers of the second kind, that involves falling factorial powers.
$$x^{n}= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}x^\underline{r}.$$
- By virtue of the reciprocity law for factorial powers, we can transform this into a formula involving rising factorial powers as follows $$\begin{align}
x^{n}&= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^r(-x)^\overline{r}\quad\Longleftrightarrow\nonumber\\
(-x)^{n}&= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^rx^\overline{r}\quad\Longleftrightarrow\nonumber\\
(-1)^nx^{n}&= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^rx^\overline{r}\quad\Longleftrightarrow\nonumber\\
x^{n}&= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^{r-n}x^\overline{r}\quad\Longleftrightarrow\nonumber\\
x^{n}&= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^{n-r}x^\overline{r}.\nonumber\\
\end{align}$$
∎
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References
Bibliography
- Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
