Files and links (1)
Preprint (Author's original)arXiv.org - Non-exclusive license to distribute, Open
Preprint
Zeckendorf family identities generalized
arXiv (Cornell University)
23 Mar 2011
Abstract
Philip Matchett Wood and Doron Zeilberger have constructed identities for the
Fibonacci numbers $f_n$ of the form $1f_n = f_n$ for all $n \geq 1$; $2f_n =
f_{n-2} + f_{n+1}$ for all $n \geq 3$; $3f_n = f_{n-2} + f_{n+2}$ for all $n
\geq 3$; $4f_n = f_{n-2} + f_{n} + f_{n+2}$ for all $n \geq 3$; ...; the
general identity in this family has the form $kf_n = \sum_{s \in S_k} f_{n+s}$
(for all sufficiently high $n$), where $S_k$ is a finite set of integers that
depends only on $k$ and contains no two consecutive integers. These identities
are generalized, replacing the left-hand side $kf_n$ by arbitrary sums of the
form $f_{n+a_1} + f_{n+a_2} + \cdots + f_{n+a_p}$ for arbitrary integers $a_1,
a_2, \ldots, a_p$. The resulting theorem is proved using the connection between
the Fibonacci numbers and the golden ratio.
Metrics
6 Record Views
Details
- Title
- Zeckendorf family identities generalized
- Creators
- Darij Grinberg
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862235604721