5,258 Results Found

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be view...

Seven infinite series involving two free variables and central binomial coefficients (in denominators) are explicitly evaluated in closed form. Several identities regarding Pell/Lucas polynomials and Fibonacci/Lucas numbers are presented as consequen...

In this work, we define higher-order Jacobsthal–Lucas quaternions with the help of higher-order Jacobsthal–Lucas numbers. We examine some identities of higher-order Jacobsthal–Lucas quaternions. We introduce their basic definitions...

This study introduces two new sequences: the incomplete Edouard and the incomplete Edouard–Lucas numbers. In addition, we establish some of the properties, identities, and recurrence relations of these sequences. The relations of these new sequ...

The hypercube is one of the best models for the network topology of a distributed system. Recently, Padovan cubes and Lucas–Padovan cubes have been introduced as new interconnection topologies. Despite their asymmetric and relatively sparse int...

We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequ...

Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Luca...

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some...

In this study, firstly the definitions and basic algebraic properties of k-Oresme and k-Oresme–Lucas sequences are given. Then, various summation formulae are derived with the help of the first and second derivatives of two polynomials with k-O...

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balanc...

The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenions and derive relationships between them.

Some new formulas related to the well-known symmetric Lucas polynomials are the primary focus of this article. Different approaches are used for establishing these formulas. A matrix approach to Lucas polynomials is followed in order to obtain some f...

This study investigates the closed forms of Lucas matrices, with a particular emphasis on the $n^{th}$ powers of the tridiagonal symmetric Toeplitz matrix $S_4(x,y)$, whose entries are associated with Lucas numbers $L_n$. The analysis extends Filipponi’s...

During the last decade, many researchers have focused on proving identities that reveal the relation between Fibonacci and Lucas numbers. Very recently, one of these identities has been generalized to the case of Fibonacci and Lucas numbers of order...

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this sy...

We have investigated new Pauli Fibonacci and Pauli Lucas quaternions by taking the components of these quaternions as Gaussian Fibonacci and Gaussian Lucas numbers, respectively. We have calculated some basic identities for these quaternions. Later,...

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with $n\ge 2$, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r m...

In this study, the $k$-Oresme and $k$-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the $k$-Oresme and $k$-Oresme-Lucas sequences are presented. In addition, we give these sequences t...

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind ${}_2{F}_1\left(z\right)$ are included in all connection coefficients for a specific z. Several ne...

In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas function...

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different...

The purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders $q\in (0,1)$ and $q\in (1,2)$. This paper also explores the fractional Pell&ndas...

By drawing on the concepts of Jacobsthal polynomials, Jacobsthal–Lucas polynomials, and hybrid numbers, this paper constructs, for the first time, a novel class of mathematical objects with recursive properties—namely, the sequences of Ja...

In this paper, we find all Lucas numbers written in the form $\overline{c\cdots cd\cdots d}$, where $\overline{c\cdots cd\cdots d}$ is the concatenation of two repdigits in base 10 with $c,d\in \{0,1,\cdots ,9\}$, $c\ne d$ and $c>0$.

Proof of Storage-Time (PoST) is the core verification mechanism for blockchain data storage, ensuring the integrity and continuous availability of data throughout the storage period. Although the current mainstream Compact Proofs of Storage-Time (cPo...

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also poin...

Optical flow algorithms offer a way to estimate motion from a sequence of images. The computation of optical flow plays a key-role in several computer vision applications, including motion detection and segmentation, frame interpolation, three-dimens...

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the $(p,q)$&ndas...

In this paper, we find all Fibonacci and Lucas numbers written in the form $2^a+3^b+5^c+7^d$ , in non-negative integers $a,b,c,d$ , with $0\le \max \{a,b,c\}\le d$ .

Norm bounds for circulant-type matrices associated with the bi-periodic Pell–Lucas sequence are examined from a symmetry-driven perspective. By incorporating alternating recurrence coefficients, the results clarify how periodicity and circulant...

In recent years, the global development community has been emphasizing blended finance approaches for economic development without taking into consideration practical implications of the Lucas Paradox, or the observation that capital does not flow fr...

There are many new land use and land cover (LULC) products emerging yet there is still a lack of in situ data for training, validation, and change detection purposes. The LUCAS (Land Use Cover Area frame Sample) survey is one of the few authoritative...

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybr...

The use of in situ references in Earth observation monitoring is a fundamental need. LUCAS (Land Use and Coverage Area frame Survey) is an activity that has performed repeated in situ surveys over Europe every three years since 2006. The dataset is u...

The Lucas balancing polynomial is linked to a family of bi-starlike functions denoted as ${\mathcal{S}}_{sc}^c(\vartheta ,\Xi \left(x\right))$, which we present and examine in this work. These functions are defined with respect to symmetric conjugate points. Coefficient estimate...

In this paper we consider a generalized bi-periodic Fibonacci $\left\{f_n\right\}$ and a generalized bi-periodic Lucas sequence $\left\{q_n\right\}$ which are respectively defined by $f_0=0$, $f_1=1$, $f_n=a{f}_{n-1}+c{f}_{n-2}$ (n is even) or $f_n=b{f}_{n-1}+c{f}_{n-2}$ (n is odd), and $q_0$...

The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method,...

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sum...

This work employs newly shifted Lucas polynomials to approximate solutions to the time-fractional Fitzhugh–Nagumo differential equation (TFFNDE) relevant to neuroscience. Novel essential formulae for the shifted Lucas polynomials are crucial fo...

This study explored the potential of the Land Use/Cover Area frame Survey (LUCAS) data for generating detailed Land Use and Land Cover (LULC) maps. Although earth observation (EO) satellites provide extensive temporal and spatial coverage, limited re...

In the contemporary paper, we introduce new subclasses of analytic and bi-univalent functions involving integral operator based upon Lucas polynomial. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functio...

In this paper, we derive the well-defined solutions to a $\theta$-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the glo...

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer $k\ge 2$ . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the $p^e$ -...

Classification of remote sensing images using machine learning models requires a large amount of training data. Collecting this data is both labor-intensive and time-consuming. In this study, the effectiveness of using pre-existing reference data on...

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric func...

Two nonlinear Duffing equations are numerically treated in this article. The nonlinear fractional-order Duffing equations and the second-order nonlinear Duffing equations are handled. Based on the collocation technique, we provide two numerical algor...

In this paper, I examine the ideas regarding image reception that can be extracted from the De altera uita, a theological treatise written by the Iberian bishop Lucas de Tui in ca. 1230. In this book, he devotes one chapter to rejecting the changes t...

Several secure image encryption systems have been researched and formed by chaotic mechanisms in current decades. This work recommends an innovative quantum color image encryption method focused on the Lucas series-based substitution box to enhance t...

In this paper, we introduce a new subclass of bi-univalent functions defined using Lucas-Balancing polynomials. For functions in each of these bi-univalent function subclasses, we derive estimates for the Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$...

Displacement is critical when it comes to the evaluation of civil structures. Large displacement can be dangerous. There are many methods that can be used to monitor structural displacements, but every method has its benefits and limitations. Lucas&n...