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


Source of Name

This entry was named for James Stirling.

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