Zeckendorf family identities generalized
Darij Grinberg

TL;DR
This paper generalizes Zeckendorf family identities for Fibonacci numbers, extending their form to arbitrary sums of Fibonacci terms and proving the results via the Fibonacci-golden ratio connection.
Contribution
It introduces a broad generalization of Fibonacci identities, allowing arbitrary sums on the left side, and provides a proof leveraging the Fibonacci-golden ratio relationship.
Findings
Generalized identities for Fibonacci numbers with arbitrary sums
Proved identities using Fibonacci and golden ratio connection
Extended previous specific identities to a wider class
Abstract
Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers of the form for all ; for all ; for all ; for all ; ...; the general identity in this family has the form (for all sufficiently high ), where is a finite set of integers that depends only on and contains no two consecutive integers. These identities are generalized, replacing the left-hand side by arbitrary sums of the form for arbitrary integers . The resulting theorem is proved using the connection between the Fibonacci numbers and the golden ratio.
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