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

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## Contents

## 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 $\displaystyle {n \brack k}$ 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.

## 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.