Definition:Stirling Numbers of the First Kind/Signed/Definition 1
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Definition
Signed Stirling numbers of the first kind are defined recursively by:
- $\map s {n, k} := \begin{cases}
\delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$
where:
- $\delta_{n k}$ is the Kronecker delta
- $n$ and $k$ are non-negative integers.
Notation
The notation $\ds {n \brack k}$ for the unsigned Stirling numbers of the first kind is that proposed by Jovan Karamata and publicised by Donald E. Knuth.
The notation $\map s {n, k}$ for the signed Stirling numbers of the first kind is similar to variants of that sometimes given for the unsigned.
Usage is inconsistent in the literature.
Also see
- Equivalence of Definitions of Unsigned Stirling Numbers of the First Kind
- Equivalence of Definitions of Signed Stirling Numbers of the First Kind
Source of Name
This entry was named for James Stirling.