# 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.

## Also see

## 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.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(44)$