Sum over k of Stirling Number of the Second Kind of n+1 with k+1 by Unsigned Stirling Number of the First Kind of k with m by -1^k-m

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Theorem

Let $m, n \in \Z_{\ge 0}$.

$\ds \sum_k {n + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} = \binom n m$

where:

$\dbinom n m$ denotes a binomial coefficient
$\ds {k \brack m}$ denotes an unsigned Stirling number of the first kind
$\ds {n + 1 \brace k + 1}$ denotes a Stirling number of the second kind.


Proof

The proof proceeds by induction on $n$.


For all $n \in \Z_{\ge 0}$, let $\map P m$ be the proposition:

$\ds \forall m \in \Z_{\ge 0}: \sum_k {n + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} = \binom n m$


Basis for the Induction

$\map P 0$ is the case:

\(\ds \) \(\) \(\ds \sum_k {0 + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m}\)
\(\ds \) \(=\) \(\ds \sum_k \delta_{1 \paren {k + 1} } {k \brack m} \paren {-1}^{k - m}\) Stirling Number of the Second Kind of 1
\(\ds \) \(=\) \(\ds \sum_k \delta_{0 k} {k \brack m} \paren {-1}^{k - m}\) Definition of Kronecker Delta
\(\ds \) \(=\) \(\ds {0 \brack m} \paren {-1}^{- m}\) All terms but where $k = 0$ vanish
\(\ds \) \(=\) \(\ds \delta_{0 m} \paren {-1}^{- m}\) Unsigned Stirling Number of the First Kind of 0
\(\ds \) \(=\) \(\ds \delta_{0 m}\) multiplier irrelevant when $m \ne 0$
\(\ds \) \(=\) \(\ds \binom 0 m\) Zero Choose n

So $\map P 0$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $\map P r$ is true, where $r \ge 1$, then it logically follows that $\map P {r + 1}$ is true.


So this is the induction hypothesis:

$\ds \sum_k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} = \binom r m$


from which it is to be shown that:

$\ds \sum_k {r + 2 \brace k + 1} {k \brack m} \paren {-1}^{k - m} = \binom {r + 1} m$


Induction Step

This is the induction step:

\(\ds \) \(\) \(\ds \sum_k {r + 2 \brace k + 1} {k \brack m} \paren {-1}^{k - m}\)
\(\ds \) \(=\) \(\ds \sum_k \paren {\paren {k + 1} {r + 1 \brace k + 1} + {r + 1 \brace k} } {k \brack m} \paren {-1}^{k - m}\) Definition of Stirling Numbers of the Second Kind
\(\ds \) \(=\) \(\ds \sum_k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \sum_k k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} + \sum_k {r + 1 \brace k} {k \brack m} \paren {-1}^{k - m}\)
\(\ds \) \(=\) \(\ds \binom r m + \sum_k k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} + \sum_k {r + 1 \brace k} {k \brack m} \paren {-1}^{k - m}\) Induction Hypothesis
\(\ds \) \(=\) \(\ds \binom r m + \sum_k k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {-1}^{r + 1 - m} \sum_k {r + 1 \brace k} {k \brack m} \paren {-1}^{r + 1 - k}\)
\(\ds \) \(=\) \(\ds \binom r m + \sum_k k {r + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} + \paren {-1}^{r + 1 - m} \delta_{m \paren {r + 1} }\) Second Inversion Formula for Stirling Numbers
\(\ds \) \(=\) \(\ds \sum_k k {r + 1 \brace k + 1} \paren {\frac 1 k {k + 1 \brack m} - \frac 1 k {k \brack m - 1} } \paren {-1}^{k - m}\) Definition of Unsigned Stirling Numbers of the First Kind
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\)
\(\ds \) \(=\) \(\ds \sum_k {r + 1 \brace k + 1} {k + 1 \brack m} \paren {-1}^{k - m} + \sum_k {r + 1 \brace k + 1} {k \brack m - 1} \paren {-1}^{k - m}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\)
\(\ds \) \(=\) \(\ds \sum_k {r + 1 \brace k + 1} {k + 1 \brack m} \paren {-1}^{k - m} + \binom r {m - 1}\) Induction Hypothesis
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\)
\(\ds \) \(=\) \(\ds \sum_k {r + 1 \brace k} {k \brack m} \paren {-1}^{k - 1 - m} + \binom r {m - 1}\) Translation of Index Variable of Summation
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\)
\(\ds \) \(=\) \(\ds \paren {-1}^{r - m} \sum_k {r + 1 \brace k} {k \brack m} \paren {-1}^{r + 1 - k} + \binom r {m - 1}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\)
\(\ds \) \(=\) \(\ds \paren {-1}^{r - m} \delta_{m \paren {r + 1} } + \binom r {m - 1} + \binom r m - \paren {-1}^{r - m} \delta_{m \paren {r + 1} }\) Second Inversion Formula for Stirling Numbers
\(\ds \) \(=\) \(\ds \binom r {m - 1} + \binom r m\) simplifying
\(\ds \) \(=\) \(\ds \binom {r + 1} m\) Pascal's Rule


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


Therefore:

$\ds \forall m, n \in \Z_{\ge 0}: \sum_k {n + 1 \brace k + 1} {k \brack m} \paren {-1}^{k - m} = \binom n m$

$\blacksquare$


Also see


Sources

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