Search Results (16)

In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By employing Binet-type formulas, recurrence relations, Chebyshev polynomial connections, and classical transformation properties of Gauss hypergeometric functions, we derive several explicit and alternative representations for the Pell and Jacobsthal numbers. These representations unify known identities, yield new closed-form expressions, and reveal deeper structural parallels between hypergeometric functions and linear recurrence sequences. The results demonstrate that hypergeometric functions provide a systematic and versatile analytical tool for studying special number sequences beyond the Fibonacci case, and they suggest potential extensions to broader families such as Horadam-type sequences and their generalizations. Full article

In this study, we introduce the generalized Repunit sequence and its hybrid quaternion extension derived from a parametric recurrence relation that preserves the base-10 structure of classical Repunit numbers. Fundamental properties of the proposed sequences, including the characteristic equation, generating function, and Binet-type formula, are systematically investigated. Several algebraic identities, such as bilinear index-reduction formulas, are established to demonstrate the internal structure and consistency of the construction. Numerical experiments and graphical analyses are conducted to examine the structural behavior of the generalized Repunit sequence and its hybrid quaternion counterpart. While the scalar Repunit sequence exhibits regular and predictable growth, the hybrid quaternion extension displays significantly higher structural complexity and variability. Density distributions, contour plots, histogram representations, and discrete variation measures confirm the presence of enhanced diffusion and local irregularity in the quaternion-based structure. These statistical, graphical, and numerical findings indicate that generalized Repunit hybrid quaternion sequences possess properties that are relevant to encoding, masking, and preprocessing mechanisms in applied mathematical and computational frameworks. However, this work does not propose a complete cryptographic algorithm, nor does it claim compliance with established cryptographic security standards such as NIST SP 800-22. The results should therefore be interpreted as pre-cryptographic indicators that motivate further research toward rigorous security evaluation, algorithmic development, and broader applications in areas such as coding theory, signal processing, and nonlinear dynamical systems. 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

In this study is introduced a novel generalization of repunit and one-zero numbers through the formulation of the generalized One-kZero. This sequence extends the classical families of repunit and one-zero numbers by establishing a unified framework in which the parameter $(k\ge 0)$ specifies the number of consecutive zeros separating two ones in the decimal representation. We introduce the new family of sequences, the generalized One-kZero numbers, and investigate some of their properties. The main purpose is to present a generalization for the recurrence relation of kind One-Zero numbers and determine some relations and properties. The reason that led us to this method is that the recurrence relation of One-Zero and Repunit numbers has a second-order difference equation as a specific case of the Horadam-type sequence. The Binet formula, generating function, sum formulas and many other relations will therefore be much easier to find. Also, some other identities that have not been found before in the particular case of One-Zero and Repunit sequences are also included in this study. 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

This study introduces an innovative approach to Horadam sequences. The aim of this paper is to investigate the Tricomplex Horadam sequence and its properties. It begins with the tricomplex ring $\mathbb{T}$ and key results related to Horadam-type sequences. The Tricomplex Horadam sequence is then defined, with a discussion of its properties, the Binet formula, and the generating function in vector form. Next, several fundamental identities, including the Tagiuri–Vajda and d’Ocagne identities, are examined, along with their implications and examples from previous Tricomplex Horadam-type sequences. Finally, the sum of terms associated with the Tricomplex Horadam sequence is presented. The research problem consists in determining properties symmetrical or analogous to the Horadam-type sequence for the Tricomplex Horadam sequence. Full article

This article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class extends the class of log-balanced sequences by including the sequences of similar growth rates, but of the opposite log-behavior. Special attention is paid to the sequences defined by two- and three-term linear recurrences with constant coefficients. For the special cases of generalized Fibonacci and Lucas sequences, we graphically illustrate the domains of their log-convexity and log-concavity. For an application, we establish the concyclicity of the points $\left({\displaystyle \frac{a_{2n}}{a_{2n+1}}},{\displaystyle \frac{1}{a_{2n+1}}}\right)$ for some classes of Horadam sequences $(a_n)$ with positive terms. Full article

This study introduces an innovative approach to Mersenne-type numbers. This paper introduces a new class of numbers, which we call Gersenne numbers. The aim of this paper is to define the Gersenne sequence and to investigate some of their properties, such as the recurrence relation, the summation formula, and the generating function. Moreover, the classical identities are derived, such as the Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne identities for Gersenne numbers. Full article

This paper proposes a numerical technique to solve the time-fractional generalized Kawahara differential equation (TFGKDE). Certain shifted Lucas polynomials are utilized as basis functions. We first establish some new formulas concerned with the introduced polynomials and then tackle the equation using a suitable collocation procedure. The integer and fractional derivatives of the shifted polynomials are used with the typical collocation method to convert the equation with its governing conditions into a system of algebraic equations. The convergence and error analysis of the proposed double expansion are rigorously investigated, demonstrating its accuracy and efficiency. Illustrative examples are provided to validate the effectiveness and applicability of the proposed algorithm. Full article

In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$, which is defined by the initial terms $r_0=0$ and $r_1=1$ and follows the Horadam recurrence relation given by $r_n=11r_{n-1}-10r_{n-2}$ for $n\ge 2$. Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring $\mathbb{T}$. We define the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$, with initial terms $t{r}_0=(0,1,11)$ and $t{r}_1=(1,11,111)$, and governed by the recurrence relation $t{r}_n=11t{r}_{n-1}-10t{r}_{n-2}$, for $n\ge 2$. This sequence is also a Horadam-type sequence but defined in the tricomplex ring $\mathbb{T}$. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet’s formula, Tagiuri–Vajda’s identity, d’Ocagne’s identity, and Catalan’s identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence. 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

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

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. Full article

Horadam sequence is a general recurrence of second order in the complex plane, depending on four complex parameters (two initial values and two recurrence coefficients). These sequences have been investigated over more than 60 years, but new properties and applications are still being discovered. Small parameter variations may dramatically impact the sequence orbits, generating numerous patterns: periodic, convergent, divergent, or dense within one dimensional curves. Here we explore Horadam sequences whose orbit is dense within a 2D region of the complex plane, while the complex argument is uniformly distributed in an interval. This enables the design of a pseudo-random number generator (PRNG) for the uniform distribution, for which we test periodicity, correlation, Monte Carlo estimation of $\pi$, and the NIST battery of tests. We then calculate the probability distribution of the radii of the sequence terms of Horadam sequences. Finally, we propose extensions of these results for generalized Horadam sequences of third order. Full article

In this paper, we consider an approach based on the elementary matrix theory. In other words, we take into account the generalized Gaussian Fibonacci numbers. In this context, we consider a general tridiagonal matrix family. Then, we obtain determinants of the matrix family via the Chebyshev polynomials. Moreover, we consider one type of tridiagonal matrix, whose determinants are Horadam hybrid polynomials, i.e., the most general form of hybrid numbers. Then, we obtain its determinants by means of the Chebyshev polynomials of the second kind. We provided several illustrative examples, as well. Full article