# Definition:Stirling Numbers of the First Kind/Signed/Definition 2

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## Contents

## Definition

**Signed Stirling numbers of the first kind** are defined as the polynomial coefficients $\map s {n, k}$ which satisfy the equation:

- $\displaystyle x^{\underline n} = \sum_k \map s {n, k} x^k$

where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.

## Notation

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