Fibonacci identities and Fibonacci pairs

TL;DR

This paper introduces Fibonacci pairs, a class of matrix pairs with entries related to Fibonacci and Lucas numbers, and systematically constructs identities by analyzing specific cases.

Contribution

It defines Fibonacci pairs of matrices and develops a systematic method to derive identities involving Fibonacci and Lucas numbers.

Findings

Construction of Fibonacci pairs of rank 2 and 3

Systematic derivation of Fibonacci identities

Matrices with entries as Fibonacci or Lucas polynomials

Abstract

A Fibonacci pair $F_s(w,x)$ of rank $s$ is a pair $s \times s$ nonsingular matrices such that $wx=xw$ and that the entries of $aw^n$ and $axw^m$ are polynomials of Fibonacci or Lucas numbers for some nonzero $a$. We construct identities systematically by the study of $F_2(w, x)$ and $F_3(w, x)$.