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## Proof: (related to "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}
q.e.d

| | | | created: 2020-04-05 19:04:11 | modified: 2020-04-05 19:06:46 | by: | references: [1112]