Fibonacci Polynomials Research Articles

Integrated Lucas and Fibonacci Polynomial Approach for Time Fractional Fast and Slow Diffusion Models

Integrated Lucas and Fibonacci Polynomial Approach for Time Fractional Fast and Slow Diffusion Models

A Family of Bi-Univalent Functions Defined by( p, q)-Derivative Operator Subordinate to a GeneralizedBivariate Fibonacci Polynomials

Making use of a generalized bivariate Fibonacci polynomials, we propose a family of normalized regular functions ψ(ζ) = ζ + d2ζ2 + d3ζ3 + · · · , which are bi-univalent in the disc {ζ ∈ C : |ζ| < 1} involving (p, q)-derivative operator. We find estimates on the coefficients |d2|, |d3| and the Fekete-Szeg¨o inequality for members of this family. New implications of the primary result as well as pertinent links to previously published findings are also provided.

On a Class of Generalized Multivariate Hermite–Humbert Polynomials via Generalized Fibonacci Polynomials

This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

A new quadra polynomial sequence

In this paper, we define a new quadra polynomial sequence by using Özkoç numbers as the coefficients. Then, we derive some properties for this polynomial sequence by the help of Fibonacci and Pell polynomials. Additionally, we attempt to define the companion matrix of this polynomial sequence.

Novel Classes on Generating Functions of the Products of (p,q)-Modified Pell Numbers with Several Bivariate Polynomials

In this paper, using the symmetrizing operator δe1e22−l, we derive new generating functions of the products of p,q-modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials.

A matrix approach to generalized Bernoulli–Fibonacci polynomials of order m and applications

A matrix approach to generalized Bernoulli–Fibonacci polynomials of order m and applications

New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation

Abstract This article is dedicated to propose a spectral solution for the non-linear Fitzhugh–Nagumo equation. The proposed solution is expressed as a double sum of basis functions that are chosen to be the convolved Fibonacci polynomials that generalize the well-known Fibonacci polynomials. In order to be able to apply the proposed collocation method, the operational matrices of derivatives of the convolved Fibonacci polynomials are introduced. The convergence and error analysis of the double expansion are carefully investigated in detail. Some new identities and inequalities regarding the convolved Fibonacci polynomials are introduced for such a study. Some numerical results, along with some comparisons, are provided. The presented results show that our proposed algorithm is efficient and accurate.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem

A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem

On Generalized Fibospinomials: Generalized Fibonacci Polynomial Spinors

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 special number sequences have also revealed a new approach to the use of spinors in mathematics and have provided a different perspective for spinor research that can be used as a source for future physics studies. The purpose of this work is to expand the generalized Fibonacci quaternion polynomials to the generalized Fibonacci polynomial spinors by associating spinors with quaternions, and to introduce and investigate a new polynomial sequence that can be used to benefit from the potential advantages of spinors in physical applications, and thus, to provide mathematical arguments, such as new polynomials, for studies using spinors and quaternions in quantum mechanics. Starting from this point of view, in this paper we introduce and investigate a new family of sequences called generalized Fibospinomials (or generalized Fibonacci polynomial spinors or Horadam polynomial spinors). Being particular cases, we use (r,s)-Fibonacci and (r,s)-Lucas polynomial spinors. We present Binet’s formulas, generating functions and the summation formulas for these polynomials. In addition, we obtain some special identities of these new sequences and matrices related to these polynomials. The importance of this study is that generalized Fibospinomials are currently the most generalized sequence in the literature when moving from Fibonacci quaternions to spinor structure, and that a wide variety of new spinor sequences can be obtained from this particular polynomial sequence.

Melham’s sums for some Lucas polynomial sequences

A Lucas polynomial sequence is a pair of generalized polynomial sequences that satisfy the Lucas recurrence relation. Special cases include Fibonacci polynomials, Lucas polynomials, and Balancing polynomials. We define the (a,b)-type Lucas polynomial sequences and prove that their Melham’s sums have some interesting divisibility properties. Results in this paper generalize the original Melham’s conjectures.

On Certain Properties of Parametric Kinds of Apostol-Type Frobenius–Euler–Fibonacci Polynomials

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program.

Polynomial Values with Integer Coefficients for The Generating Functions of Fibonacci Polynomials

Polynomial Values with Integer Coefficients for The Generating Functions of Fibonacci Polynomials

Catalan Identities for Generalized Fibonacci Polynomials

Catalan Identities for Generalized Fibonacci Polynomials

Spectral properties of a functional binomial matrix

In this article, we consider the definition of the Fibonacci polynomial sequence with the second-order linear recurrence relation, where coefficients and initial conditions depend on the variable $t$. And then, we introduce the functional binomial matrix depending on the coefficients of the second-order linear recurrence relation. In the following, we study the spectral properties of the functional binomial matrix using the Fibonacci polynomial sequence and we obtain a diagonal decomposition for it using the Vandermunde matrix. Finally, by applying some linear algebra tools we obtain a number of combinatorial identities involving the Fibonacci polynomial sequence.

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.

Subdomain collocation method based on successive integration technique for solving delay differential equations

The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.

Solving neutral delay differential equations using least square method based on successive integration technique

The main objective of this work is to propose the Least square method (LSM) using successive integration technique for solving Neutral delay differential equations (NDDEs). Continuous LSM and Discrete LSM have been presented by adopting different orthogonal polynomials as weighted basis functions. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear NDDEs have been provided to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems.

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

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 polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained.

On hyper (r, q)-Fibonacci polynomials

Abstract Related to generalized arithmetic triangle, we introduce the hyper (r, q)-Fibonacci polynomials as the sum of these elements along a finite ray starting from a specific point, which generalize the hyper-Fibonacci polynomials. We give generating function, recurrence relations and we show some properties whose application allows us to extend the notion of Cassini determinant and to study some ratios. Moreover, we derive a connection between these polynomials and the incomplete (r, q)-Fibonacci polynomials defined in this paper.