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

## Definition

Unsigned Stirling numbers of the first kind are defined as the polynomial coefficients $\displaystyle {n \brack k}$ which satisfy the equation:

$\displaystyle x^{\underline n} = \sum_k \left({-1}\right)^{n - k} {n \brack k} x^k$

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.

## 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.