Search Results (34)

In this article, we present new theoretical findings on specific polynomials that generalize the concept of telephone numbers, namely, Telephone polynomials (TelPs). Several new formulas are developed, including expressions for higher-order derivatives, repeated integrals, and moment formulas of TelPs. Moreover, we derive explicit connections between the derivatives of TelPs and the two classes of symmetric and non-symmetric polynomials, producing many formulas between these polynomials and several celebrated polynomials such as Hermite, Laguerre, Jacobi, Fibonacci, Lucas, Bernoulli, and Euler polynomials. The inverse formulas are also obtained, expressing the derivatives of well-known polynomial families in terms of TelPs. Furthermore, some novel linearization formulas (LFs) with some classes of polynomials are established. Finally, some new definite and indefinite integrals of TelPs are established using some of the developed relations. Full article

The Horadam sequence $H_n(a,b;p,q)$ unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials ($v_1,v_2,v_3$) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices $kn$ through a parameterized grammar $G_k$. Using the modified grammar $\widetilde{G_k}$, we derive convolution formulas involving the weighting term ${(-q)}^{n-i}$. Furthermore, applying the proposed framework to $(p,q)$-Fibonacci and $(p,q)$-Lucas sequences, we derive explicit convolution formulas with parameters $(p,q)$. The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers. Full article

The Horadam sequence ${\left\{H_n(a,b;p,q)\right\}}_{n⩾0}$ has been widely studied in combinatorics and number theory. In this paper, we find that the context-free grammar $G=\{x\to px+y,y\to qx\}$ can be used to generate Horadam sequences. Using this grammar, we deduce several identities, including Cassini-like identities. Moreover, we investigate the relationship between two distinct Horadam sequences $H_n(a,b;p,q)$ and $H_n(c,d;p,q)$ with $(a,b)\ne (c,d)$ and provide an approach to derive identities, which can be illustrated by the Fibonacci and Lucas sequences as well as the two kinds of Chebyshev polynomials. Full article

In this study, we define a new family of Gaussian polynomials, called Gaussian Chebyshev polynomials, by extending classical Chebyshev polynomials into the complex domain. These polynomials are characterized by second-order linear recurrence relations, and their connections with the Chebyshev polynomials are established. We also examine properties such as Binet-type formulas and generating functions. Moreover, we characterize some relationships between Gaussian and classical Chebyshev polynomials for the first and second kinds. We obtain some well-known theorems, such as Cassini, Catalan, and d’Ocagne’s theorems, for the first and second kinds. Furthermore, we present important connections among four types of these new polynomials. In the proofs of our results, we utilize the symmetric and antisymmetric properties of the Chebyshev polynomials. Finally, it is shown that Gaussian Chebyshev polynomials are closely related to well-known special sequences such as the Fibonacci, Lucas, Gaussian Fibonacci, and Gaussian Lucas numbers for some specific values of variables. Full article

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences. Full article

In this study, firstly the definitions and basic algebraic properties of k-Oresme and k-Oresme–Lucas sequences are given. Then, various summation formulae are derived with the help of the first and second derivatives of two polynomials with k-Oresme and k-Oresme–Lucas number coefficients. The main aim of this study is to establish the relations between the generalized Fibonacci and generalized Lucas sequences and the k-Oresme and k-Oresme–Lucas sequences, respectively. These connections allow us to obtain different combinatorial identities of these sequences using the characteristic equation of the k-Oresme and k-Oresme–Lucas sequences. In this way, the discovered combinatorial identities reveal the arithmetic and structural symmetries in the sequences, through the regularities and recurring patterns observed in the algebraic structures of the considered number sequences. The results obtained in this study enable the development of new symmetric approaches in areas such as numerical analysis, cryptography and optimization algorithms, and the algebraic relations derived in this study can contribute to the solution of different problems in disciplines such as mathematical modelling and theoretical physics. Full article

Two nonlinear Duffing equations are numerically treated in this article. The nonlinear fractional-order Duffing equations and the second-order nonlinear Duffing equations are handled. Based on the collocation technique, we provide two numerical algorithms. To achieve this goal, a new family of basis functions is built by combining the sets of Fibonacci and Lucas polynomials. Several new formulae for these polynomials are developed. The operational matrices of integer and fractional derivatives of these polynomials, as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert nonlinear Duffing equations into algebraic systems of equations by forcing the equation to hold at certain collocation points. To numerically handle the resultant nonlinear systems, one can use symbolic algebra solvers or Newton’s approach. Some particular inequalities are proved to investigate the convergence analysis. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions by utilizing Fibonacci–Lucas polynomials as basis functions. Full article

This article introduces new polynomials that extend the standard Leonardo numbers, generalizing Fibonacci and Lucas polynomials. A new power form representation is developed for these polynomials, which is crucial for deriving further formulas. This article also presents two connection formulas linking these generalized polynomials to the Fibonacci and Lucas polynomials, as well as several identities involving some generalized and specific Leonardo numbers. Additionally, new product formulas involving the generalized Leonardo polynomials with the Fibonacci and Lucas polynomials are provided, along with computations of definite integrals based on the derived formulas. 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

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article

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. Full article

In this paper, using the symmetrizing operator ${\delta }_{e_1{e}_2}^{2-l}$, we derive new generating functions of the products of $\left(p,q\right)$-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. Full article

The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources. Full article

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. Full article