Generalized Lucas Numbers and Relations with Generalized Fibonacci Numbers

TL;DR

This paper introduces a new matrix-based generalization of Lucas numbers using generalized Lucas polynomials, explores their properties, and establishes relations with generalized Fibonacci numbers, including formulas and combinatorial representations.

Contribution

It presents a novel matrix representation of generalized Lucas numbers and derives their properties and relations with generalized Fibonacci numbers, including Binet formulas.

Findings

Established properties of the new generalized Lucas numbers.

Derived relations between generalized Lucas and Fibonacci numbers.

Provided Binet formula and combinatorial representations.

Abstract

In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.