165 Results Found

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relation...

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain...

A generalization of the well-known Fibonacci sequence is the $k-$Fibonacci sequence whose first k terms are $0,\dots ,0,1$ and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i...

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain ge...

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the...

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...

Carlitz solved some Diophantine equations on Fibonacci or Lucas numbers. We extend his results to the sequence of generalized Fibonacci and Lucas numbers. In this paper, we solve the Diophantine equations of the form ...

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...

In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find a...

In this paper, we prove that $F_{22}=17711$ is the largest Fibonacci number whose decimal expansion is of the form $ab\dots bc\dots c$ . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and...

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...

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a...

This paper establishes symmetric inequalities for the reciprocal sums of Fibonacci numbers. By using elementary methods, recurrence properties, Cassini’s identity, and the Fibonacci–Lucas connection, we deduce the precise bounds for these...

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...

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,...

The sequence of the k-generalized Fibonacci numbers ${(F_n^{(k)})}_n$ is defined by the recurrence $F_n^{(k)}={\sum }_{j=1}^k{F}_{n-j}^{(k)}$ beginning with the k terms $0,\dots ,0,1$ . In this paper,...

We introduce a new subclass $\mathbf{J}{\Sigma }^{\eta ,\delta ,\mu }\left(\tilde{p}\right)$ of bi-univalent functions, defined by shell-like curves connected with Fibonacci numbers. Our main results in this paper include estimates of the Taylor–Maclaurin coefficients $\left|a_2\right|$...

This paper concerns the properties of the generalized bi-periodic Fibonacci numbers $\left\{G_n\right\}$ generated from the recurrence relation: $G_n=a{G}_{n-1}+G_{n-2}$ (n is even) or $G_n=b{G}_{n-1}+G_{n-2}$ (n is odd). We derive general identities for the recip...

Let ${(t_n^{\left(r\right)})}_{n\ge 0}$ be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence $t_n^{\left(r\right)}=t_{n-1}^{\left(r\right)}+\cdots +t_{n-r}^{\left(r\right)}$ for $n\ge r$, with initial values $t_0^{\left(r\right)}=0$ and $t_i^{\left(r\right)}=1$, for all $1\le i\le r$. In 2002, Grossman and Luca searched for terms o...

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...

Emphasising their connection with shell-like star-like curves, this work investigates a new subclass of star-like functions defined by $\mathfrak{q}$-Fibonacci numbers and $\mathfrak{q}$-polynomials. We study the geometric and analytic properties of this subclass, including t...

In this investigation, by using the Komatu integral operator, we introduce the new class of bi-univalent functions based on the rule of subordination. Moreover, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the gen...

For $r\ge 2$ and $a\ge 1$ integers, let ${\left(t_n^{(r,a)}\right)}_{n\ge 1}$ be the sequence of the $(r,a)$-generalized Fibonacci numbers which is defined by the recurrence $t_n^{(r,a)}=t_{n-1}^{(r,a)}+\cdots +t_{n-r}^{(r,a)}$ for $n>r$, with initial values $t_i^{(r,a)}=1$, for all $i\in [1,r-1]$ and $t_r^{(r,a)}=a$. In this...

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order gener...

Pseudorandom number and bit sequence generators are widely used in cybersecurity, measurement, and other technology fields. A special place among such generators is occupied by additive Fibonacci generators (AFG). By itself, such a generator is not c...

In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class $\mathcal{A}$ of normalized analytic functions, where class $\mathcal{A}$ is invariant...

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recur...

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences....

Let $k\ge 1$ be an integer and denote ${\left(F_{k,n}\right)}_n$ as the k-Fibonacci sequence whose terms satisfy the recurrence relation $F_{k,n}=k{F}_{k,n-1}+F_{k,n-2}$ , with initial conditions $F_k$...

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ having initial terms $f_0=0$, and $f_1=1$ and recurrence relation $f_n=f_{n-1}+f_{n-2}(n\ge 2)$...

The aim of this paper is to discuss the emergence of recursive thinking through the famous problem posed by Fibonacci regarding the growth of the rabbit population. This paper qualitatively analyzes and discusses the semiotic aspects raised by the st...

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can...

In this paper, we consider an approach based on the elementary matrix theory. In other words, we take into account the generalized Gaussian Fibonacci numbers. In this context, we consider a general tridiagonal matrix family. Then, we obtain determina...

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 $0\le \max \{a,b,c\}\le d$ .

Random numbers are widely employed in cryptography and security applications. If the generation process is weak, the whole chain of security can be compromised: these weaknesses could be exploited by an attacker to retrieve the information, breaking...

Here, we describe the bicomplex $\left(p,q\right)$-Fibonacci numbers and the bicomplex $\left(p,q\right)$-Fibonacci quaternions based on these numbers to show that bicomplex numbers are not defined the same as bicomplex quaternions. Then, we give some of their equations...

The aim of this paper is to investigate the solution of the following difference equation $z_{n+1}={(p_n)}^{-1},n\in {\mathbb{N}}_0,{\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ where $p_n=a+b{z}_n+c{z}_{n-1}{z}_n$ with the parameters a, b, c and the initial values $z_{-1},z_0$ are nonzero quaternions such...

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the go...

In this paper, with the help of the finite operators and Fibonacci numbers, we define a new family of quaternions whose components are the Fibonacci finite operator numbers. We also provide some properties of these types of quaternions. Moreover, we...

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for t...

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomi...

In this paper we introduce and study $(2,k)$-distance Fibonacci polynomials which are natural extensions of $(2,k)$-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we p...

In the present paper, we first study the Gaussian Leonardo numbers and Gaussian Leonardo hybrid numbers. We give some new results for the Gaussian Leonardo numbers, including relations with the Gaussian Fibonacci and Gaussian Lucas numbers, and also...

This paper presents a comprehensive survey of the generalization of hybrid numbers and hybrid polynomials, particularly in the fields of mathematics and physics. In this paper, by using higher-order generalized Fibonacci polynomials, we introduce hig...

Spinors are important objects in physics, which have found their place more and more after the discovery that particles have an intrinsic angular momentum shape and Cartan’s mathematical expression of this situation. Recent studies using specia...

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, we define the gener...

Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. A hyperbolic complex structure is frequently used in both pure mathematics and numerous areas of physics. In this paper we introduce a special kind of spacelike hybrid number...

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 this paper, we introduce a new quadruple number sequence by means of Leonardo numbers, which we call ordered Leonardo quadruple numbers. We determine the properties of ordered Leonardo quadruple numbers including relations with Leonardo, Fibonacci...

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, t...