Generalized Fibonacci Polynomials Research Articles

On Higher-Order Generalized Fibonacci Hybrinomials: New Properties, Recurrence Relations and Matrix Representations

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 higher-order generalized Fibonacci hybrid polynomials called higher-order generalized Fibonacci hybrinomials. We obtain some special cases and algebraic properties of the higher-order generalized Fibonacci hybrinomials, such as the recurrence relation, generating function, exponential generating function, Binet formula, Vajda’s identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. We also present three different matrices whose components are higher-order generalized Fibonacci hybrinomials, higher-order generalized Fibonacci polynomials and Lucas polynomials. By using these matrices, we obtain some identities related to these newly established hybrinomials.

\(q\)-Analogue of the Generalized Fibonacci and Lucas Polynomials

In this article, we define \(q\)-generalized Fibonacci polynomials and \(q\)-generalized Lucas polynomials using \(q\)-binomial coefficient and obtain their recursive properties. In addition, we introduce generalized \(q\)-Fibonacci matrix and generalized \(q\)-Lucas matrix, then we derive their basic identities. We define \((k,q,t)\)-symmetric generalized Fibonacci matrix and \((k,q,t)\)-symmetric generalized Lucas matrix, then we give the Cholesky factorization of these matrices. Finally, we give determinantal and permanental representations of these new polynomial sequences.

On the separation of the roots of the generalized Fibonacci polynomial

In this paper we prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values.

On symmetric Tetranacci polynomials in mathematics and physics

In this manuscript, we introduce (symmetric) Tetranacci polynomials ξ j as a twofold generalization of ordinary Tetranacci numbers, considering both non unity coefficients and generic initial values. We derive a complete closed form expression for any ξ j with the key feature of a decomposition in terms of generalized Fibonacci polynomials. For suitable conditions, ξ j can be understood as the superposition of standing waves. The issue of Tetranacci polynomials originated from their application in condensed matter physics. We explicitly demonstrate the approach for the spectrum, eigenvectors, Green’s functions and transmission probability for an atomic tight binding chain exhibiting both nearest and next nearest neighbor processes. We demonstrate that in topological trivial models, complex wavevectors can form bulk states as a result of the open boundary conditions. We describe how effective next nearest neighbor bonding is engineered in state of the art theory/experiment exploiting onsite degrees of freedom and close range hopping. We argue about experimental tune ability and on-demand complex wavevectors.

McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].

Optimal study on fractional fascioliasis disease model based on generalized Fibonacci polynomials

Fascioliasis is a liver fluke disease in which food and water are the transmitting agents. The disease is caused by a genus of Fasciola, parasitic Trematoda. The genus Fasciola includes two species of Fasciola gigantica and Fasciola hepatica. The study of this disease is of particular importance as the infected cases are globally growing, especially in endemic areas. In this paper, a fractional‐order mathematical model is presented for fascioliasis disease. The optimization method is an effective choice in solving mathematical models of epidemic diseases. The study aims at presenting a new set of generalized Fibonacci polynomials as basic functions and some innovative operational matrices of derivatives in the Caputo sense to solve the fascioliasis disease model. The numerical experiments reveal better results and simulations show effectiveness of the presented method.

Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials

In this paper, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain a new explicit formula for the generalized Humbert–Hermite polynomials. We provide determinantal representations for the ratio of two differentiable functions. We obtain a recursive relation for the generalized Humbert–Hermite polynomials. As a practice, we derive an alternative recursive relation for generalized Humbert–Hermite polynomials via the Hessenberg determinant. Finally, we derive several families of multilinear and multilateral generating functions for the generalized Humbert–Hermite‐type polynomials and other polynomials which are mentioned in this paper. Our results also include many well‐known polynomials in the literature.

AN EXTENDED GENERALIZED FIBONACCI POLYNOMIAL BASED CODING METHOD WITH ERROR DETECTION AND CORRECTION

A Fibonacci coding method is introduced using Extended Generalized Fibonacci Polynomials in this paper. A new square matrix Qⁿm (a, b), the nᵗʰ power of Qm(a, b) of order m × m is defined whose elements are based on extended Generalized Fibonacci Polynomial. Matrix Qⁿ m (a, b) for integer x ≥ 1, a ≥ 1 and b ≥ 1 is considered as the encoding matrix and a matrix Q−ⁿm (a, b) is considered as decoding matrix. An error-detection and error-correction method is also defined in Extended Generalized Fibonacci polynomials.

Triangular numbers and generalized fibonacci polynomial

Abstract In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A052529 that has strong relationships with triangular numbers.

On Generalized Fibonacci Polynomials: Horadam Polynomials

In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and we deal with, in detail, two special cases which we call them $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functions, Simson's formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices associated with these sequences. Finally, we present several expressions and combinatorial results of the generalized Fibonacci polynomials.

Spectral Collocation Algorithm for Solving Fractional Volterra-Fredholm Integro-Differential Equations via Generalized Fibonacci Polynomials

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

Square numbers, square pyramidal numbers, and generalized Fibonacci polynomials

In this paper, we derive two interesting formulas for square and square pyramidal numbers. We focus on the linear recurrence relation with constant coefficients for square and square pyramidal numbers. Then we deal with the relationship between generalized Fibonacci polynomials and these numbers. Also, we give some determinant representations of these numbers.

McKay matrix for indecomposable module of finite representation type Hopf algebra

Let H be a Hopf algebra of finite representation type. For the grouplike elements and skew-primitive elements in H, we prove some general results about the eigenvalues and eigenvectors of the McKay matrix in the framework of Green ring. As an example, we investigate the McKay matrix WV of Taft algebra for tensoring with an indecomposable -module Furthermore, we compute the characteristic polynomial of WV in terms of a kind of generalized Fibonacci polynomial, and give some eigenvector(s) of each eigenvalue for WV by relating to the generalized Fibonacci polynomial. At last, we determine the eigenspaces of all eigenvalues.

On a class of generalized Humbert-Hermite polynomials via generalized Fibonacci polynomials

A unified presentation of a class of Humbert's polynomials in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinney, Horadam-Pethe, Djordjevi${\acute{c}}$, Gould, Milovanovi${\acute{c}}$ and Djordjevi${\acute{c}}$, Pathan and Khan polynomials and many not so called 'named' polynomials has inspired the present paper. We define here generalized Humbert-Hermite polynomials of two variables. Several expansions of Humbert-Hermite polynomials, Hermite-Gegenbaurer (or ultraspherical) polynomials

Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

<abstract><p>In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.</p></abstract>

The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix

We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix.
 These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial
 sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which
 generalize the previous results in references.

SEVERAL GENERATING FUNCTIONS OF GENERALIZED FIBONACCI POLYNOMIALS

In this paper, we obtain the generating functions up to third order of generalized Fibonacci polynomials defined by R. Florez. Also we obtain several generating functions of several famous polynomials and sequences as particular cases.

A note on the coefficient array of a generalized Fibonacci polynomial

A particular version of a Fibonacci polynomial is presented and the coefficients are tabulated to bring out some of their number theory properties with known results. Generalizations of the fundamental and primordial Lucas sequences are used in the proofs.

On some properties of generalized Fibonacci polynomials

On some properties of generalized Fibonacci polynomials