Search Results (42)

In this paper, we study a generalization of Padovan numbers. We define a generating function and matrix generators for the generalized Padovan sequence. Moreover, using graph methods and a special family of generalized Padovan sequences, we derive a multinomial formula for generalized Padovan numbers. We also prove some identities that generalize known formulae for the classical Padovan numbers. Full article

In the general linear Lie algebra of continuous linear transformations in n dimensions, we show that unequal Abelian scaling transformations on the components of a vector can stabilize the system information in the presence of Markov component transformations on the vector, which, alone, would lead to increasing entropy. The more interesting results follow from seeking Diophantine (integer) solutions, with the result that the system can be stabilized with constant information for each of a set of entropy rates ($k=\mathrm{1,2},3,\dots$ ). The first of these—the simplest—where $k=1$, results in the Fibonacci sequence, with information determined by the olden mean, and Fibonacci interpolating functions. Other interesting results include the fact that a new set of higher order generalized Fibonacci sequences, functions, golden means, and geometric patterns emerges for $k=2, 3, \dots$ Specifically, we define the $k$^th order golden mean as ${\Phi }_k=k/2+\sqrt{{(k/2)}^2+1}$ for $k =1, 2, 3, \dots .$. One can easily observe that one can form a right triangle with sides of 1 and $k/2$ and that this will give a hypotenuse of $\sqrt{{(k/2)}^2+1}$. Thus, the sum of the $k/2$ side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden means for any value of k relative to the third side with a value of unity. The sequential powers of the matrix $(k^2+1,k,k,1)$ for any integer value of $k$ provide a generalized Fibonacci sequence. Also, using the general equation expressed as ${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{(k/2)}^2+1}$ for $k =\mathrm{1,2},3, \dots ,$ one can easily prove that ${\Phi }_k=k+1/{\Phi }_k$ which is a generalization of the familiar equation expressed as $\Phi =1+1/\Phi$. We suggest that one could look for these new ratios and patterns in nature, with the possibility that all of these systems are connected with the retention of information in the presence of increasing entropy. Thus, we show that two components of the general linear Lie algebra ($GL(n,\mathbb{R})$), acting simultaneously with certain parameters, can stabilize the information content of a vector over time. Full article

This paper presents a modification of an image encryption algorithm combining chaos and the Fibonacci matrix by integrating artificial intelligence via a Generative Pre-Trained Transformer (GPT). The goal is to improve the robustness of the algorithm by dynamically adapting the parameters of the chaotic system and generating cryptographic keys based on image characteristics. The proposed methodology includes two main innovations: the implementation of GPT for automated generation of the initial parameters of the chaotic system, which allows for greater variability and security in encryption, and the use of GPT for dynamic determination of the Fibonacci Q-matrix, which provides additional complexity and increased resistance to attacks. The method is realized in the MATLAB (R2023a) environment through integration with OpenAI API and the MATLAB–Python interface for requesting GPT models. The efficiency and reliability of the modified algorithm are compared with those of standard chaotic encryption algorithms, and its robustness to various cryptographic attacks is also studied. Full article

The rapid development of network communication technology has led to an increased focus on the security of image storage and transmission in multimedia information. This paper proposes an enhanced image security communication scheme based on Fibonacci interleaved diffusion and non-degenerate chaotic system to address the inadequacy of current image encryption technology. The scheme utilizes a hash function to extract the hash characteristic values of the plaintext image, generating initial perturbation keys to drive the chaotic system to generate initial pseudo-random sequences. Subsequently, the input image is subjected to a light scrambling process at the bit level. The Q matrix generated by the Fibonacci sequence is then employed to diffuse the obtained intermediate cipher image. The final ciphertext image is then generated by random direction confusion. Throughout the encryption process, plaintext correlation mechanisms are employed. Consequently, due to the feedback loop of the plaintext, this algorithm is capable of resisting known-plaintext attacks and chosen-plaintext attacks. Theoretical analysis and empirical results demonstrate that the algorithm fulfils the cryptographic requirements of confusion, diffusion, and avalanche effects, while also exhibiting a robust password space and excellent numerical statistical properties. Consequently, the security enhancement mechanism based on Fibonacci interleaved diffusion and non-degenerate chaotic system proposed in this paper effectively enhances the algorithm’s resistance to cryptographic attacks. Full article

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities $A_n^2-A_{n+1}{A}_{n-1}=k^{-2n}$, $A_n^2-A_n{A}_{n-1}+{\displaystyle \frac{1}{k^2}}{A}_{n-1}^2=k^{-2n}$, and $B_n^2-B_n{B}_{n-1}+{\displaystyle \frac{1}{k^2}}{B}_{n-1}^2=-(k^2-4)k^{-2n}$, and some generalizations such as $B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2$, $A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$, and more were derived, where $m,n,t\in \mathbb{Z}$ and $t\ne n$. In addition to this, the solution pairs of the algebraic equations $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$, $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$, and $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$ are presented, where $A_p$ and $B_p$ are k-Oresme and k-Oresme–Lucas numbers, respectively. Full article

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article

This paper proposes a novel image encryption and decryption scheme, called Square Block Division-Fibonacci (SBD-Fibonacci), which dynamically partitions any input image into optimally sized square blocks to enable efficient encryption without resizing or distortion. The proposed encryption scheme can dynamically adapt to the image dimensions and ensure compatibility with images of varying and high resolutions, while it serves as a yardstick for any symmetric-key image encryption algorithm. An optimization model, combined with the Lagrange Four-Square theorem, minimizes trivial block sizes, strengthening the encryption structure. Encryption keys are generated using the direct sum of generalized Fibonacci matrices, ensuring key matrix invertibility and strong diffusion properties and security levels. Experimental results on widely-used benchmark images and a comparative analysis against State-of-the-Art encryption algorithms demonstrate that SBD-Fibonacci achieves high entropy, strong resistance to differential and statistical attacks, and efficient runtime performance—even for large images. Full article

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 foundational work by examining distinct cases of ordered pairs $(x,y)$, thereby determining the precise conditions under which $S_4(x,y)$ qualifies as a Fibonacci–Lucas matrix. Furthermore, it identifies specific conditions under which $S_4(x,y)$ can be classified as any Fibonacci–Lucas matrix. These findings contribute to the theoretical framework of Fibonacci–Lucas matrices and provide novel insights into their structural properties. Full article

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 generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple. Full article

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations. Full article

Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author’s background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the method of undetermined coefficients and several others. Among these, the generating function approach is particularly prevalent in enumerative combinatorics due to its versatility and widespread use. The primary objective of this work is to introduce an alternative representation grounded in the theory of Riordan arrays. This representation provides a general formula expressed in terms of the vectors of constants and initial conditions associated with any recurrence relation of a given order, offering a new perspective on the structure of such sequences. Full article

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. Full article

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 properties of the eigenvalues of their related $2\times 2$ matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated $2\times 2$ matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given. Full article

The main focus in this manuscript is to find a numerical solution of a dengue fever disease model by using the Fibonacci wavelet method. The operational matrix of integration has been obtained using Fibonacci wavelets. The proposed method is called Fibonacci wavelet collocation method (FWCM). This biological model has been transformed into a system of nonlinear algebraic equations by using the Fibonacci wavelet collocation scheme. Afterward, this system has been solved by using the Newton–Raphson method. Finally, we provide evidence that our results are better than those obtained by various current approaches, both numerically and graphically, demonstrating the method’s accuracy and efficiency. Full article

In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators for these sequences and offered a graphical representation to clarify their structure. Furthermore, we demonstrated how these generalizations can be applied to obtain the Padovan and Narayana sequences for specific parameter values. Full article