Second Inversion Formula for Stirling Numbers

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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:

\(\displaystyle \sum_k \left\{ {0 \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{0 - k}\) \(=\) \(\displaystyle \sum_k \delta_{k 0} \left[{k \atop m}\right] \left({-1}\right)^{-k}\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle \left[{0 \atop m}\right]\) as all other terms vanish by $\delta_{k 0}$
\(\displaystyle \) \(=\) \(\displaystyle \delta_{m 0}\) Definition of Unsigned Stirling Numbers of the First Kind

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:


\(\displaystyle \) \(\) \(\displaystyle \sum_k \left\{ { {r + 1} \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_k \left({k \left\{ {r \atop k}\right\} + \left\{ {r \atop k - 1}\right\} }\right) \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}\) Definition of Stirling Numbers of the Second Kind
\(\displaystyle \) \(=\) \(\displaystyle \sum_k k \left\{ {r \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k} + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_k k \left\{ {r \atop k}\right\} \frac 1 k \left({\left[{k + 1 \atop m}\right] - \left[{k \atop m - 1}\right]}\right) \left({-1}\right)^{r + 1 - k}\) Definition of Unsigned Stirling Numbers of the First Kind
\(\displaystyle \) \(\) \(\displaystyle + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(\) \(\displaystyle + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(\) \(\displaystyle + \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k + 1}\) Translation of Index Variable of Summation
\(\displaystyle \) \(=\) \(\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(\) \(\displaystyle - \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r - k}\) Simplification
\(\displaystyle \) \(=\) \(\displaystyle \delta_{\left({m - 1}\right) r}\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \delta_{m \left({r + 1}\right)}\) Definition of Kronecker Delta


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


Therefore:

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

$\blacksquare$


Also see


Sources