# Definition:Stirling Numbers of the First Kind/Unsigned

## Definition

In the below:

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

### Definition 1

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

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

### Definition 2

**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 $x$.

## Stirling's Triangle of the First Kind (Unsigned)

- $\begin{array}{r|rrrrrrrrrr} n & {n \brack 0} & {n \brack 1} & {n \brack 2} & {n \brack 3} & {n \brack 4} & {n \brack 5} & {n \brack 6} & {n \brack 7} & {n \brack 8} & {n \brack 9} \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 2 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 0 & 6 & 11 & 6 & 1 & 0 & 0 & 0 & 0 & 0 \\ 5 & 0 & 24 & 50 & 35 & 10 & 1 & 0 & 0 & 0 & 0 \\ 6 & 0 & 120 & 274 & 225 & 85 & 15 & 1 & 0 & 0 & 0 \\ 7 & 0 & 720 & 1764 & 1624 & 735 & 175 & 21 & 1 & 0 & 0 \\ 8 & 0 & 5040 & 13068 & 13132 & 6769 & 1960 & 322 & 28 & 1 & 0 \\ 9 & 0 & 40320 & 109584 & 118124 & 67284 & 22449 & 4536 & 546 & 36 & 1 \\ \end{array}$

## Extension to Complex Numbers

Donald E. Knuth, in his *The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed.* of $1997$, suggests an extension of the **unsigned Stirling numbers of the first kind** $\displaystyle {r \brack r - m}$ to the real and complex numbers.

However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.

## Also defined as

Some sources do not introduce the signed Stirling numbers of the first Kind, and therefore refer to these as just the **Stirling numbers of the first kind**.

## 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.

## Examples

### $5$th Falling Factorial

- $x^{\underline 5} = x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x$

and so:

- $\dbinom x 5 = \dfrac 1 {120} \paren {x^5 - 10 x^4 + 35 x^3 - 50 x^2 + 24 x}$

## Also see

- Definition:Signed Stirling Numbers of the First Kind
- Definition:Stirling Numbers of the Second Kind
- Definition:Pascal's Triangle

- Results about
**Stirling numbers (of both the first and second kind)**can be found here.

## 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