Generalized Fibonacci Polynomials Research Articles

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

Distance Fibonacci Polynomials

In this paper, we introduce a new kind of generalized Fibonacci polynomials in the distance sense. We give a direct formula, a generating function and matrix generators for these polynomials. Moreover, we present a graph interpretation of these polynomials, their connections with Pascal’s triangle and we prove some identities for them.

Chords of an Ellipse, Lucas Polynomials, and Cubic Equations

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse generalizing a classic problem about circles. We give a brief history of the circle problem an account of Price’s ellipse proof and a reorganized proof with some new ideas designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano’s solution of the cubic equation and Newton’s theorem on power sums and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. We also develop additional connections that surface along the way; e.g., we give a parallel interpretation of generalized Fibonacci polynomials and we show that Cardano’s method can be used to write down the roots of the Lucas polynomials.

On the spectrum of finite, rooted homogeneous trees

In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials and produce a limiting distribution for the eigenvalues as the tree depth goes to infinity. We indicate how these results can be extended to periodic branching patterns and also provide a generalization to higher order simplicial complexes.

A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.

Genelleştirilmiş Fibonacci ve Lucas polinomlarıyla birlikte sağ devirsel matrisler ve kodlama teorisi

In the present paper, we consider two new coding algorithms by means of right circulant matrices with elements generalized Fibonacci and Lucas polynomials. To that end, we study basic properties of right circulant matrices using generalized Fibonacci polynomials, generalized Lucas polynomials and geometric sequences.

Generalized Fibonacci Operational Collocation Approach for Fractional Initial Value Problems

A numerical algorithm for solving multi-term fractional differential equations (FDEs) is established herein. We established and validated an operational matrix of fractional derivatives of the generalized Fibonacci polynomials (GFPs). The proposed numerical algorithm is mainly built on applying the collocation method to reduce the FDEs with its initial conditions into a system of algebraic equations in the unknown expansion coefficients. Output of the numerical results asserted that our developed algorithm is applicable, efficient and accurate.

Spectral Tau Algorithm for Certain Coupled System of Fractional Differential Equations via Generalized Fibonacci Polynomial Sequence

This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.

Rational convolution roots of isobaric polynomials

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under the convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first and second kind. Hence, we call them Stirling operators. In order to construct matrix representations of the roots of GFPs, we use Stirling operators of the first kind. We give explicit examples to show how Stirling operators of the second kind appear in low degree cases for the WIP-roots. As a consequence of the matrix construction we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.

Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for Fp, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

Generalized Humbert polynomials via generalized Fibonacci polynomials

In this paper, we define the generalized (p, q)-Fibonacci polynomials un, m(x) and generalized (p, q)-Lucas polynomials vn, m(x), and further introduce the generalized Humbert polynomials un,m(r)(x) as the convolutions of un, m(x). We give many expressions, expansions, recurrence relations and differential recurrence relations of un,m(r)(x), and study the matrices and determinants related to the polynomials un, m(x), vn, m(x) and un,m(r)(x). Finally, we present an algebraic interpretation for the generalized Humbert polynomials un,m(r)(x). It can be found that various well-known polynomials are special cases of un, m(x), vn, m(x) or un,m(r)(x). Therefore, by introducing the general polynomials un, m(x), vn, m(x) and un,m(r)(x), we have a unified approach to dealing with many special polynomials in the literature.

Generalized Fibonacci Polynomials and some Identities

Generalized Fibonacci Polynomials and some Identities

Generalized Fibonacci Polynomials

In this study, we present generalized Fibonacci polynomials. We have used their Binet’s formula and generating function to derive the identities. The proofs of the main theorems are based on special functions, simple algebra and give several interesting properties involving them.

HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials

The focus of this paper is to study the HOMFLY polynomial of $(2,n)$-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of $ (2,n) $-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental properties. We also show that the HOMFLY polynomial of $ (2,n) $-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial. We finally give the matrix representations and prove important identities, which are similar to the Fibonacci identities, for the our generalized Fibonacci polynomial and the HOMFLY polynomial of $ (2,n) $-torus link.

Generalizations of Fibonacci polynomials and free linear groups

The aim of this paper is to relate some generalized Fibonacci polynomials to the problem of knowing when a group (or semigroup) generated by some linear transformations is free.

Generalized Fibonacci Polynomials and Fibonomial Coefficients

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials \({\{n\}}\)in variables s,t given by \({\{0\} = 0, \{1\} = 1}\), and \({\{n \} = s\{n - 1 \} +t\{n - 2 \}}\) for \({{n \geq 2}}\) . The latter are defined by \({\left\{\begin{array}{ll} k \end{array}\right\} = \{ n \}! / (\{ k \}!\{ n - k \}!)}\) where \({{\{n \}! = \{1 \}\{2 \}\cdots\{n \}}}\). These quotients are also polynomials in s, t and specializations give the ordinary binomial coefficients, the Fibonomial coefficients, and the q-binomial coefficients. We present some of their fundamental properties, including a more general recursion for \({\{n\}}\) , an analogue of the binomial theorem, a new proof of the Euler- Cassini identity in this setting with applications to estimation of tails of series, and valuations when s and t take on integral values. We also study a corresponding analogue of the Catalan numbers. Conjectures and open problems are scattered throughout the paper.

Generalized Fibonacci-Like Polynomials and Some Identities

The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Lucas Polynomials is introduced and defined by with and. some basic identities established and derived by standard methods. Keywords : Fibonacci Polynomials, Lucas Polynomials, Generalized Fibonacci Polynomials MSC: 2000: 11B37, 11B39

The convolution ring of arithmetic functions and symmetric polynomials

Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas poly- nomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to produce a theory of logarithms and exponentials within arithmetic functions giving another proof of the fact that the group of multiplicative functions under convolution product is isomorphic to the group of additive functions under addition. The hyperbolic trigonometric functions are constructed from the EXP operator, again, in the usual way. The usual hyperbolic trigonometric identities hold. We exhibit new structure and identities in the isobaric ring. Given a monic polynomial, its infinite companion matrix can be embedded in the group of weighted isobaric polynomials. The derivative of the monic polynomial and its compan- ion matrix give us the different matrix and the infinite different matrix. The determinant of the different matrix is the discriminate of the monic polynomial up to sign. In fact, the LOG operating on the infinite companion matrix is the infinite different matrix. We prove that an arithmetic function is locally representable if an only if it is a multiplicative function. An arithmetic function is both locally and globally representable if it is trivially globally represented. Keyword. Arithmetic functions; Multiplicative functions; Additive functions; Sym- metric polynomials; Isobaric polynomials; Generalized Fibonacci polynomials; Generalized Lucas polynomials MSC. 11B39; 11B75; 11N99; 11P99; 05E05