# Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1

## Definition

Unsigned Stirling numbers of the first kind are defined recursively by:

$\displaystyle {n \brack k} := \begin{cases} \delta_{n k} & : k = 0 \text { or } n = 0 \\ & \\ \displaystyle {n - 1 \brack k - 1} + \paren {n - 1} {n - 1 \brack k} & : \text{otherwise} \\ \end{cases}$

where:

$\delta_{n k}$ is the Kronecker delta
$n$ and $k$ are non-negative integers.

## Notation

The notation $\displaystyle {n \brack k}$ 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.

## Source of Name

This entry was named for James Stirling.

## Technical Note

The $\LaTeX$ code for $\displaystyle {n \brack k}$ is \displaystyle {n \brack k} .

The braces around the n \brack k are important.

The \displaystyle is needed to create the symbol in its proper house display style.