A weighted extension of Fibonacci numbers

TL;DR

This paper introduces a weighted extension of Fibonacci numbers, generalizes multiple identities, and explores an elliptic extension related to $q$-Fibonacci polynomials, with proofs mainly combinatorial and telescoping.

Contribution

It presents a novel weighted Fibonacci extension, generalizes existing identities, and introduces an elliptic extension linked to $q$-Fibonacci polynomials, expanding theoretical understanding.

Findings

Extended Fibonacci identities with weights

Introduced elliptic extension related to $q$-Fibonacci polynomials

Provided combinatorial and telescoping proofs

Abstract

We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the $q$ case, by Garrett. Some identities are proved by telescoping.