Definition:Stirling Numbers of the First Kind/Unsigned

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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:

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


Also see

  • 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