Investigation of Determinants of Fibonacci-Hessenberg-Lorentz Matrices and Special Number Sequences

TL;DR

This paper introduces a new class of matrices called Fibonacci-Hessenberg-Lorentz matrices, constructed via Fibonacci and Lorentz matrices, to explore their determinants and potential to generate or generalize special number sequences.

Contribution

It develops a novel matrix type combining Fibonacci, Hessenberg, and Lorentz matrices and investigates their determinants for generating and generalizing special number sequences.

Findings

Determinants of these matrices can produce known and new number sequences.

A generalized formula for sequence terms based on matrix parameters is proposed.

The matrices reveal connections between matrix properties and classical number sequences.

Abstract

The research aims to construct a new type of matrix called the Fibonacci-Hessenberg-Lorentz matrix by multiplying Fibonacci-Hessenberg matrices with Lorentz matrix multiplication. The study will start by examining the properties of Hessenberg and tridiagonal matrices and then focus on developing the Fibonacci-Hessenberg matrix using Fibonacci sequences. By multiplication it with a Lorentz matrix multiplication, the resulting matrix, the Fibonacci-Hessenberg-Lorentz matrix, will be analyzed to obtain special number sequences through its determinants for n>=1. The primary objective is to explore whether the determinants of these matrices can generate new or known number sequences, where the elements are expressed as functions of the matrix parameters. Furthermore, the research will attempt to generalize these sequences of using Fibonacci numbers to establish a generalized formula for their terms. Ultimately, the goal is to derive a mathematical representation that connects the characteristics of the newly defined matrices to well-known special sequences in mathematics.