arXiv subjects

Rigoberto Flórez

Publications and source records attributed to Rigoberto Flórez.

At least 19 recordsLinked to original sources

arXiv2026

Holonomic Sequences Arising from Laplace Transforms of Generalized Second Order Sequences

We study the FiboLaplace and LucasLaplace sequences obtained by applying the Laplace transform to generalized Fibonacci-type and Lucas-type polynomials. We relate these families and show that, for each fixed transform parameter, the FiboLaplace sequence is holonomic. We give a combinatorial interpretation in terms of weighted colored tilings, leading to recurrence relations, identities, and a triangular refinement by the number of dominoes. We also derive two continued-fraction expansions and obtain a second combinatorial interpretation in terms of generalized Motzkin paths. These results extend earlier work of Givens and Moll.

math.CO↗

arXiv2025

Zeros and Orthogonality of generalized Fibonacci polynomials

This paper analyzes the concept of orthogonality in second-order polynomial sequences that have Binet formula similar to that of the Fibonacci and Lucas numbers, referred to as Generalized Fibonacci Polynomials (GFP). We give a technique to find roots of the GFP. As a corollary of this result, we give an alternative proof of a special case of Favard's Theorem. The general case of Favard's Theorem guarantees that there is a measure to determine whether a sequence of second-order polynomials is orthogonal or not. However, the theorem does not provide an explicit such measure. Our special case gives both the explicit measure and the relationship between the second-order recurrence and orthogonality, demonstrating whether the GFP polynomials are orthogonal or not. This allows us to classify which of familiar GFPs are orthogonal and which are not. Some familiar orthogonal polynomials include the Fermat, Fermat-Lucas, both types of Chebyshev polynomials, both types of Morgan-Voyce polynomials, and Vieta and Vieta-Lucas polynomials. However, we prove that the Fibonacci, Lucas, Pell, and Pell-Lucas sequences are not orthogonal. In Section \ref{sectionrw}, we give a brief description of discrete--time and continuous--time Morkov chains with special emphasis on birth-and-death stochastic processes. We find sufficient conditions on the polynomial's coefficients under which a given family of orthogonal polynomial induces a Markov chain. These families of orthogonal polynomials include Chebyshev polynomials of first kind and Fermat-Lucas. In the final section, we highlight some connections between orthogonal polynomials and Markov processes. These relations are not new but seem to have been somewhat forgotten. We do so to draw the attention of researchers in the orthogonal polynomial and probability communities for further collaboration.

math.CO↗

arXiv2024

Fibonacci--Theodorus Spiral and its properties

Inspired by the ancient spiral constructed by the greek philosopher Theodorus which is based on concatenated right triangles, we have created a spiral. In this spiral, called \emph{Fibonacci--Theodorus}, the sides of the triangles have lengths corresponding to Fibonacci numbers. Towards the end of the paper, we present a generalized method applicable to second-order recurrence relations. Our exploration of the Fibonacci--Theodorus spiral aims to address a variety of questions, showcasing its unique properties and behaviors. For example, we study topics such as area, perimeter, and angles. Notably, we establish a relationship between the ratio of two consecutive areas and the golden ratio, a pattern that extends to angles sharing a common vertex. Furthermore, we present some asymptotic results. For instance, we demonstrate that the sum of the first $n$ areas comprising the spiral approaches a multiple of the sum of the initial $n$ Fibonacci numbers. Moreover, we provide a sequence of open problems related to all spiral worked in this paper. Finally, in his work Hahn, Hahn observed a potential connection between the golden ratio and the ratio of areas between spines of lengths $\sqrt{F_{n+1}}$ and $\sqrt{F_{n+2}-1}$ and the areas between spines of lengths $\sqrt{F_{n}}$ and $\sqrt{F_{n+1}-1}$ in the Theodorus spiral. However, no formal proof has been provided in his work. In this paper, we provide a proof for Hahn's conjecture.

math.GM↗

arXiv2024

Projective Rectangles: The Graph of Lines

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique structure. The main construction of projective rectangles, stated in a previous paper, gives rectangles whose graph of lines is a known strongly regular bilinear forms graph. That fact leads to a proof that the main construction does produce projective rectangles, and also gives a new representation of bilinear forms graphs. We conclude by mentioning a few simple graph properties, such as the chromatic number, which is not known, and a partial geometry obtained from the graph.

math.CO↗

arXiv2023

Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions

A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane that starts at the origin, ends on the $x$-axis, and consists of the same number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a subpath of the form $DU$. A Dyck path is called \emph{restricted $d$-Dyck} if the difference between any two consecutive valleys is at least $d$ (right-hand side minus left-hand side) or if it has at most one valley. In this paper we give some connections between restricted $d$-Dyck paths and both, the non-crossing partitions of $[n]$ and some subfamilies of polyominoes. We also give generating functions to count several aspects of these combinatorial objects.

math.CO↗

arXiv2023

Activity from matroids to rooted trees and beyond

The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colored labeled rooted trees of activity \textsf{0}.

math.CO↗

arXiv2022

Primes and composites in the determinant Hosoya triangle

In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{determinant Hosoya triangle} which we denote by ${\mathcal H}$. We discuss the divisibility properties of the above numbers and their primality. We give a small sieve of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. We prove that ${\mathcal H}$ has arbitrarily large neighbourhoods of composite entries. Finally we present an abundance of data indicating a very high density of primes in ${\mathcal H}$.

math.NT↗

arXiv2022

The Geometry of some Fibonacci Identities in the Hosoya Triangle

The \emph{Hosoya triangle} is a triangular array where every entry is a product of two Fibonacci numbers. We use the geometry of this triangle to find new identities related to Fibonacci numbers. We give geometric interpretation for some well-known identities of Fibonacci numbers. For instance, the Cassini identity and the Catalan identity. We also extend some identities that hold in the Pascal triangle to the Hosoya triangle. For instance, the hockey stick extends from binomials to products of Fibonacci numbers and the rhombus property extends a binomial identity from the Pascal triangle to an identity of products of Fibonacci numbers in the Hosoya triangle.

math.NT↗

arXiv2022

Honeycombs in the Pascal triangle and beyond

In this paper we present a geometric approach to discovering some known and some new identities using triangular arrays. Our main aim is to demonstrate how to use the geometric patterns (by Carlitz), in the Pascal and Hosoya triangles to rediscover some classical identities and integer sequences. Therefore, we use new techniques in classical settings which then provide a new perspective in undergraduate research.

math.HO↗

arXiv2021

Restricted Dyck Paths on Valleys Sequence

In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of two consecutive valleys (from left to right) is at least $d$, we say that $P$ is a restricted $d$-Dyck path. The \emph{area} of a Dyck path is the sum of the absolute values of $y$-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of $d$-Dyck paths. We give a bivariate generating function to count the number of the $d$-Dyck paths with respect to the the semi-length and number of peaks. After that, we analyze in detail the case $d=-1$. Among other things, we give both, the generating function and a recursive relation for the total area.

math.CO↗

arXiv2020

Fibonacci identities from Jordan Identities

In this paper, we connect two well established theories, the Fibonacci numbers and the Jordan algebras. We give a series of matrices, from literature, used to obtain recurrence relations of second-order and polynomial sequences. We also give some identities known in special Jordan Algebras. The matrices play a bridge role between both theories. The mentioned matrices connect both areas of mathematics, special Jordan algebras and recurrence relations, to obtain new identities and classic identities in Fibonacci numbers, Lucas numbers, Pell numbers, binomial transform, tribonacci numbers, and polynomial sequences among others. The list of identities in this paper contains just a few examples of many that the reader can find using this technique.

math.NT↗

arXiv2020

Families of Integral Cographs within a Triangular Arrays

The \emph{determinant Hosoya triangle}, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle $\bmod \,2$ gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral. Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either $d$-regular graphs with $d=2,4,6,\dots $ or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.

math.CO↗

arXiv2019

Identities for the generalized Fibonacci polynomial

A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and Lucas numbers to Fibonacci type and Lucas type polynomials. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following polynomial sequences: Fibonacci with Lucas, Pell with Pell-Lucas, Fermat with Fermat-Lucas, both types of Chebyshev polynomials, Jacobsthal with Jacobsthal-Lucas and both types of Morgan-Voyce.

math.NT↗

arXiv2019

Characterization of the strong divisibility property for generalized Fibonacci polynomials

It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also giveformulas to calculate the $\gcd$ of those polynomials that do not satisfy the strong divisibility property. Note. This paper is now published in INTEGERS.

math.NT↗

arXiv2018

Matrices in the Hosoya triangle

In this paper we use well-known results from linear algebra as tools to explore some properties of products of Fibonacci numbers. Specifically, we explore the behavior of the eigenvalues, eigenvectors, characteristic polynomials, determinants, and the norm of non-symmetric matrices embedded in the Hosoya triangle. We discovered that most of these objects either embed again in the Hosoya triangle or they give rise to Fibonacci identities. We also study the nature of these matrices when their entries are taken $\bmod$ $2$. As a result, we found an infinite family of non-connected graphs. Each graph in this family has a complete graph with loops attached to each of its vertices as a component and the other components are isolated vertices. The Hosoya triangle allowed us to show the beauty of both, the algebra and geometry.

math.CO↗

arXiv2018

The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials

A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{discriminant} of a polynomial $p$ is the resultant of $p$ and its derivative. We study the resultant, the discriminant, and the derivatives of Fibonacci-type polynomials and Lucas-type polynomials as well combinations of those two types. As a corollary we give explicit formulas for the resultant, the discriminant, and the derivative for the known polynomials mentioned above.

math.NT↗

arXiv2018

Biased Graphs. VI. Synthetic Geometry

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $Ω$ has two natural matroids, the frame matroid $G(Ω)$, and the lift matroid $L(Ω)$, and their extensions the full frame matroid $G^{{}^{{}_{{}_\bullet}}\!}(Ω)$ and the extended (or complete) lift matroid $L_0(Ω)$. In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry.

math.CO↗

arXiv2017

The Projective Planarity Question for Matroids of $3$-Nets and Biased Graphs

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian. The criteria depend on the embeddability of a quasigroup associated to the graph into the additive or multiplicative loop of a ternary coordinate ring for the plane. A 3-node biased graph is equivalent to an abstract partial 3-net; thus, we have a new algebraic criterion for an abstract 3-net to be realized in a non-Desarguesian projective plane. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle, in turn, is equivalent to an isostrophe class of quasigroups, which is equivalent to a $3$-net. Much is not known about embedding a quasigroup into a ternary ring, so we do not say our criteria are definitive. For instance, it is not even known whether there is a finite quasigroup that cannot be embedded in any finite ternary ring. If there is, then there is a finite rank-3 matroid (of the corresponding biased expansion) that cannot be embedded in any finite projective plane---a presently unsolved problem.

math.CO↗