Zeckendorf family identities generalized

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.