Second Inversion Formula for Stirling Numbers

Theorem
For all $m, n \in \Z_{\ge 0}$:


 * $\displaystyle \sum_k \left\{ {n \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{n - k} = \delta_{m n}$

where:
 * $\displaystyle \left\{ {n \atop k}\right\}$ denotes a Stirling number of the second kind
 * $\displaystyle \left[{k \atop m}\right]$ denotes an unsigned Stirling number of the first kind
 * $\delta_{m n}$ denotes the Kronecker delta.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \forall m \in \Z_{\ge 0}: \sum_k \left\{ {n \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{n - k} = \delta_{m n}$

Basis for the Induction
$P \left({0}\right)$ is the case:

Thus $P \left({0}\right)$ has been shown to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({r}\right)$ is true for all $r \ge 0$, then it logically follows that $P \left({r + 1}\right)$ is true.

So this is the induction hypothesis:
 * $\displaystyle \forall m \in \Z_{\ge 0}: \sum_k \left\{ {r \atop k}\right\}\left[{k \atop m}\right] \left({-1}\right)^{r - k} = \delta_{m r}$

from which it is to be shown that:
 * $\displaystyle \forall m \in \Z_{\ge 0}: \sum_k \left\{ { {r + 1} \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k} = \delta_{m \left({r + 1}\right)}$

Induction Step
This is the induction step:

So $P \left({r}\right) \implies P \left({r + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall m, n \in \Z_{\ge 0}: \sum_k \left\{ {n \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{n - k} = \delta_{m n}$

Also see

 * First Inversion Formula for Stirling Numbers