THE BIVARIATE (COMPLEX) FIBONACCI AND LUCAS POLYNOMIALS - AN HISTORICAL INVESTIGATION WITH THE - MAPLE´S HELP

Oct 15, 2026

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Research investigates Fibonacci and Lucas sequences as enduring mathematical models that generated many specializations and generalizations, particularly from the 1960s onward. By introducing one or two variables, the study develops bivariate Lucas polynomials (BLP) and bivariate Fibonacci polynomials (BFP), emphasizing inductive reasoning and a historically grounded scientific setting supported by technology. Maple CAS commands and algebraic operations support comparisons of Lucas polynomial properties against classic Fibonacci models.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages25

Reading Time1 h 3 min

AccessFree to download

Introduction

Motivation from historical studies

The Bivariate (Complex) Fibonacci and (Complex) Lucas Polynomials

Periodicals and research interest

Polynomial sequence definitions

Generalized bivariate Fibonacci polynomials

THE BIVARIATE (COMPLEX) FIBONACCI AND LUCAS POLYNOMIALS: AN HISTORICAL INVESTIGATION WITH THE MAPLE´ S HELP Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino Abstract: The current research around the Fibonacci´s and Lucas´ sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixthies. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it´s polinomial representations. Therefore, with the introduction of one or two variables, we begin to discuss the family of the Bivariate Lucas Polynomias (BLP) and the Bivariate Fibonacci Polynomials (BFP) . On the other hand, since it ´s representation requires enormous employment of a large algebraic notational system, we explore some particular properties in order to convince the reader about an inductive reasoning that produces a meaning and produces an environment of scientific and historical investigation supported by the technology. Finally, throughout the work we bring several figures that represent some examples of commands and algebraic operations with the CAS Maple that allow to compare properties of the Lucas´polynomials, taking as a reference the classic of Fibonacci´s model that still serves as inspiration for several current studies in Mathematics. Key words: Lucas Sequence, Fibonacci´s polynomials, Historical investigation, CAS Maple. 1. Introduction Undoubtedly, the Fibonacci sequence preserves a character of interest and, at the same time, of mystery, around the numerical properties of a sequence that is originated from a problem related to the infinite reproduction of pairs of rabbits. On the other hand, in several books of History of Mathematics in Brazil and in the other countries (Eves, 1969; Gullberg, 1997; Herz, 1998; Huntley, 1970; Vajda, 1989), we appreciate a naive that usually emphasizes eminently basic and trivial properties related to this sequence. This type of approach can provide a narrow and incongruent understanding of the Fibonacci sequence, mainly about it ´s current evolutionary stage. On the other hand, from the work of the mathematicians François Édouard Anatole Lucas (1842 – 1891) and Gabriel Lamé (1795 – 1870), we observe a progressive return by the interest of the study of numerical sequences and their properties. Thus, we highlight the following set of the numerical sequences: 􀁞1,1, 2,3,5,8,13, 21,34,55,89,144, 233,377,610,987,1597, 2584, 4181,6765,10946, , fn , 􀁠 ; 􀁞1,3, 4, 7,11,18, 29, 47, 76,123,199,322,521,843,1364, 2.207,3571,5778,9349,15127,, Ln 􀁠 ; 􀁞1, 2,5,12, 29,70,168, 408,, , Pn , 􀁠 ; fn􀀎1 􀀠 fn 􀀎 fn􀀐1 ; Ln􀀎1 􀀠 Ln 􀀎 Ln􀀐1 ; Pn􀀎1 􀀠 2Pn􀀐1 􀀎 Ln􀀐2 , n 􀁴 1 . We observed that the initial values are indicated by f1 􀀠 1, f2 􀀠 1, L1 􀀠, L2 􀀠 3, P1 􀀠 1, P2 􀀠 2 . The Fibonacci polynomials were first studied in 1883 by Belgian mathematician Eugene Charles Catalan (1814 - 1894) and German mathematician Ernest Erich Jacobsthal (1881 - 1965) . Thus, Catalan defined the following family of Fibonacci polynomial functions as follows. Definition 1: We will call the Fibonacci Polynomial Sequence-SPF, the set of polynomial functions described by the recurrence relation f1 (x) 􀀠 1, f2 (x) 􀀠 x, fn (x) 􀀠 x 􀂘 fn􀀐1(x) 􀀎 fn􀀐2 (x) , n 􀁴 1 . Undoubtedly, through the previous definition, we can perceive the generalization process that initially occurred with the Fibonacci sequence and, gradually, after a few decades, began to be registered in several other generalized sequence models as well. Thus, in the following section, we will study therecursive relations arising from the introduction of two real variables, even when we consider the introduction of an imaginary unit. Such mathematical definitions are relatively recent and reflect an evolutionary mathematical and epistemological process of the Fibonacci and Lucas´ sequence. 2. The Bivariate (Complex) Fibonacci and

FAQs

What problem does the paper address regarding Fibonacci and Lucas sequences?

It explores how Fibonacci and Lucas sequences evolved into generalized polynomial models, especially bivariate (complex) polynomial representations.

How does the paper introduce bivariate polynomials?

It discusses introducing one or two variables into the sequence frameworks, leading to families of bivariate Lucas polynomials and bivariate Fibonacci polynomials.

What role does Maple CAS play in the work?

The paper uses Maple CAS commands and algebraic operations to generate examples and compare properties of Lucas polynomials with reference to classic Fibonacci models.

Oct 15, 2026

Uploaded by Rizky