Definition:Stirling Numbers of the First Kind
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Definition
Stirling numbers of the first kind come in two forms.
In the below:
- $\delta_{n k}$ is the Kronecker delta
- $n$ and $k$ are non-negative integers.
Unsigned Stirling Numbers of the First Kind
Unsigned Stirling numbers of the first kind are defined recursively by:
- $\ds {n \brack k} := \begin{cases} \delta_{n k} & : k = 0 \text { or } n = 0 \\ & \\ \ds {n - 1 \brack k - 1} + \paren {n - 1} {n - 1 \brack k} & : \text{otherwise} \\ \end{cases}$
Signed Stirling Numbers of the First Kind
Signed Stirling numbers of the first kind are defined recursively by:
- $\map s {n, k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$
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.
Examples
$5$th Falling Factorial
- $x^{\underline 5} = x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x$
and so:
- $\dbinom x 5 = \dfrac 1 {120} \paren {x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x}$
Also see
- Particular Values of Unsigned Stirling Numbers of the First Kind
- Particular Values of Signed Stirling Numbers of the First Kind
- Results about Stirling numbers (of both the first and second kind) can be found here.
Source of Name
This entry was named for James Stirling.