Category:Definitions/Stirling Numbers

This category contains definitions related to Stirling Numbers.
Related results can be found in Category:Stirling Numbers.

Stirling numbers come in various forms.

In the below:

$\delta_{n k}$ is the Kronecker delta

$n$ and $k$ are non-negative integers.

Unsigned Stirling Numbers of the First Kind

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

$\ds {n \brack k} := \begin{cases}

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

Signed Stirling Numbers of the First Kind

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}$

Stirling Numbers of the Second Kind

Stirling numbers of the second kind are defined recursively by:

$\ds {n \brace k} := \begin{cases}

\delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \ds {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$