Definition:Stirling Numbers of the First Kind/Signed

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Definition

In the below:

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

Definition 1

Signed Stirling numbers of the first kind are defined recursively by:

$\map s {n, k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$


Definition 2

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 $x$.


Stirling's Triangle of the First Kind (Signed)

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


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.