A new Fibonacci identity and its associated summation identities

TL;DR

This paper introduces a novel Fibonacci identity that unifies several classical identities and leads to new summation formulas, enhancing the theoretical understanding of Fibonacci number relationships.

Contribution

The paper presents a new Fibonacci identity that encompasses known identities and derives new binomial and summation identities, including a generalization of Halton's identity.

Findings

Unified several classical Fibonacci identities

Derived new binomial and summation identities

Generalized Halton's Fibonacci identity

Abstract

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also derive several binomial and ordinary summation identities arising from this identity; in particular we obtain a generalization of Halton's general Fibonacci identity.