Definition:Stirling Numbers of the First Kind/Signed/Definition 2
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Definition
Signed Stirling numbers of the first kind are defined as the polynomial coefficients $\map s {n, k}$ which satisfy the equation:
- $\ds x^{\underline n} = \sum_k \map s {n, k} x^k$
where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.
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.