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Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation
-
H. W. Gould
Published/Copyright:
November 25, 2009
Abstract
It is well known that the Bell numbers 
have exponential generating function 
, which satisfies the differential equation 
. In this paper, we investigate certain sequences 
whose exponential generating functions satisfy a modified form of the above differential equation, namely, the functional differential equation 
. For the main result of this paper, we show that when a = –1 and b ∈ ℝ, the sequence obeys the simple second-order linear recurrence G(n + 2) = bG(n + 1) – G(n). The proof is based on a well-known binomial series inversion formula.
Keywords.: Bell numbers; functional differential equations; Rao Uppuluri–Carpenter numbers; Fibonacci numbers
Received: 2008-10-05
Accepted: 2009-05-02
Published Online: 2009-11-25
Published in Print: 2009-November
© de Gruyter 2009
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Keywords for this article
Bell numbers;
functional differential equations;
Rao Uppuluri–Carpenter numbers;
Fibonacci numbers
Articles in the same Issue
- An Analogue of the Erdős–Ginzburg–Ziv Theorem for Quadratic Symmetric Polynomials
- Counting Determinants of Fibonacci–Hessenberg Matrices Using LU Factorizations
- A Short Proof of a Series Evaluation in Terms of Harmonic Numbers
- Generalization of an Identity Involving the Generalized Fibonacci Numbers and Its Applications
- Inductive Methods and Zero-Sum Free Sequences
- On the Number of Zero-Sum Subsequences of Restricted Size
- On the Average Asymptotic Behavior of a Certain Type of Sequence of Integers
- On the Kernel of the Coprime Graph of Integers
- Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation
- Balanced Subset Sums in Dense Sets of Integers
- Some Results for Generalized Harmonic Numbers
