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.
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. $$x^\overline{n}= \sum_{r=1}^n\left[\begin{array}{c}n\\r\end{array}\right]x^r$$
and
$$x^{n}= \sum_{r=1}^n\left\{\begin{array}{c}n\\r\end{array}\right\}(-1)^{n-k}x^\overline{r}$$
for all natural numbers $n\ge 1.$
Proofs: 1