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:
- $\ds {n \brace k} := \begin{cases}
\delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \ds {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$
Pages in category "Definitions/Stirling Numbers"
The following 25 pages are in this category, out of 25 total.
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- Definition:Signed Stirling Numbers of the First Kind
- Definition:Stirling Numbers
- Definition:Stirling Numbers of the First Kind
- Definition:Stirling Numbers of the First Kind/Notation
- Definition:Stirling Numbers of the First Kind/Signed
- Definition:Stirling Numbers of the First Kind/Signed/Definition 1
- Definition:Stirling Numbers of the First Kind/Signed/Definition 2
- Definition:Stirling Numbers of the First Kind/Unsigned
- Definition:Stirling Numbers of the First Kind/Unsigned/Complex Numbers
- Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1
- Definition:Stirling Numbers of the First Kind/Unsigned/Definition 2
- Definition:Stirling Numbers of the Second Kind
- Definition:Stirling Numbers of the Second Kind/Complex Numbers
- Definition:Stirling Numbers of the Second Kind/Definition 1
- Definition:Stirling Numbers of the Second Kind/Definition 2
- Definition:Stirling Numbers of the Second Kind/Notation
- Definition:Stirling Numbers of the Second Kind/Technical Note
- Definition:Stirling Numbers/Historical Note
- Definition:Stirling Numbers/Karamata Notation
- Definition:Stirling's Triangle of the First Kind (Signed)
- Definition:Stirling's Triangle of the First Kind (Unsigned)
- Definition:Stirling's Triangle of the Second Kind
- Definition:Stirling's Triangles