A recurrence related to the Bell numbers (Q2898391)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: A recurrence related to the Bell numbers |
scientific article; zbMATH DE number 6054413
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A recurrence related to the Bell numbers |
scientific article; zbMATH DE number 6054413 |
Statements
11 July 2012
0 references
Bell number
0 references
Stirling number
0 references
recurrence
0 references
combinatorial proof
0 references
0.9055824
0 references
0.90389365
0 references
0.89270645
0 references
0.8887044
0 references
0 references
A recurrence related to the Bell numbers (English)
0 references
The paper considers sequences \(C_n(a,b,c,d)\) for parameters \(a,b,c,d\) that satisfy the recurrence relation \( C_n(a,b,c,d)=abC_{n-1}(a,b,c,d)+cC_{n-1}(a+d,b,c,d)\) for \(n\geq 1\), with initial condition \(C_0(a,b,c,d)=1\). (The setting \(a=0, b=c=d=1\) yields the Bell numbers, and several related sequences can be obtained with other settings.) The paper solves the recurrence above in two ways, using exponential generating functions and counting arguments. As an application, a new proof is shown for the \(q\)-Stirling number formula of Carlitz.
0 references
