Signed Stirling Number of the First Kind of 1
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Theorem
- $\map s {1, n} = \delta_{1 n}$
where:
- $\map s {1, n}$ denotes a signed Stirling number of the first kind
- $\delta_{1 n}$ is the Kronecker delta.
Proof
| \(\ds \map s {1, n}\) | \(=\) | \(\ds \map s {0, n - 1} - 0 \times \map s {0, n}\) | Definition of Signed Stirling Numbers of the First Kind | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map s {0, n - 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \delta_{0 \paren {n - 1} }\) | Definition of Signed Stirling Numbers of the First Kind | |||||||||||
| \(\ds \) | \(=\) | \(\ds \delta_{1 n}\) | $0 = n - 1 \iff 1 = n$ |
$\blacksquare$