Infinite series about reciprocal products of Fibonacci and Lucas numbers are systematically examined. This is done by incorporating the telescoping method, Cassini-like identities and partial fractions, as well as the induction principle. Numerous cl...
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain...
We extend an algebraic identity of Anastase and Díaz-Barrero (2022) and apply our results to deduce various formulas for sums and series involving (among others) Fibonacci and Lucas numbers, Bernoulli polynomials, and the Riemann zeta function.
We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan–Fibonacci relations. Then, we focus on the general properties of t...
For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup
In this part of a series of two papers, we extend the theorems discussed in Part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, polylogarithms and Fibona...
In this paper, we develop two new theorems relating to the series of floor and ceiling functions. We then use these two theorems to develop more than forty distinct novel results. Furthermore, we provide specific cases for the theorems and corollarie...
We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which...
In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also poin...
Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Luca...
In this research, we explored the potential of using a research-based teaching and learning sequence to promote pupils’ engagement in practices that are coherent with those of real world mathematical and scientific activity. This STEM (Science, Techn...
In this paper, the localization of optical interface edge-states is analyzed for a dielectric material-based quasi-periodic photonic crystal (QPhC) structure. The design comprises a bilayer PhC structure, where layers are arranged in a Fibonacci conf...
In this paper, we introduce a novel class of Fubini–Fibonacci–Appell polynomials within the framework of Golden calculus. Utilizing generating function techniques and fibonomial convolution methods, we establish several structural propert...
The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is...
In this paper, we introduce the Appell–Hermite–Fibonacci polynomials within the framework of Fibonomial calculus, combining the Appell structure with a Hermite-type deformation governed by Fibonacci coefficients. The family is defined thr...
This paper introduces and investigates a family of non-symmetric Fibonacci polynomials (NSFPs) characterized by three parameters, generalizing well-known variants. We develop a series of structural formulas for these polynomials that elucidate their...
Fibonacci-spaced defect resonators were analytically investigated by cavity coupling, which exhibited a series of well-defined optical modes in fractals. The analytic model can be used to predict the output performance of microcavity lasers based on...
Seven infinite series involving two free variables and central binomial coefficients (in denominators) are explicitly evaluated in closed form. Several identities regarding Pell/Lucas polynomials and Fibonacci/Lucas numbers are presented as consequen...
In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientifi...
This paper is part of a series that describes the Fibonacci icosagrid quasicrystal (FIG) and its relation to the
This paper introduces novel subfamilies of analytic and bi-univalent functions in
Reversible data hiding (RDH) is an active area of research in the field of information security. In RDH, a secret can be embedded inside a cover medium. Unlike other data-hiding schemes, RDH becomes important in applications that demand recovery of t...
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 usefu...
Quasicrystals have attracted a growing interest in material science because of their unique properties and applications. Proper determination of the atomic structure is important in designing a useful application of these materials, for which a diffi...
In this work, we present a new way of studying the mathematical structure of the genetic code. This study relies on the use of mathematical computations involving five Fibonacci-like sequences; a few of their “seeds” or “initial con...
This work presents a review of the main topologies of switched capacitors (SCs) used in DC-DC power conversion. Initially, the basic configurations are analyzed, that is, voltage doubler, series-parallel, Dickson, Fibonacci, and ladder. Some aspects...
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...
This article presents several new analytical results for a modified class of Bernoulli polynomials, namely, the Bernoulli polynomials of the second kind (BPs2). The paper mainly develops new connection and inverse connection formulas between the firs...
This study develops an effective numerical method for addressing the time-fractional gas dynamics equation formulated with the Caputo time-fractional derivative. Novel basis functions are utilized, formulated as particular generalized Fibonacci polyn...
This paper proposes a new meta-heuristic optimization algorithm, the crown growth optimizer (CGO), inspired by the tree crown growth process. CGO innovatively combines global search and local optimization strategies by simulating the growing, sprouti...
Physical roots, exemplifications and consequences of periodic and aperiodic ordering (represented by Fibonacci series) in biological systems are discussed. The physical and biological roots and role of symmetry and asymmetry appearing in biological p...
Most crops accumulate above-ground biomass (AGB) through photosynthesis, inspiring the development of the Photosynthetic Accumulation Model (PAM) and Simplified PAM (SPAM). Both models estimate AGB based on time-series optical vegetation indices (VIs...
A rigorous method for obtaining the diffraction patterns of quasicrystals is presented. Diffraction patterns are an essential analytical tool in the study of quasicrystals, since they can be used to determine their photonic resonances. Previous metho...
Objective: This study aims to discover the most current trends, impacts, and challenges of using IoT devices and smart technologies in oral health. Method: A modified systematic mapping method was used to generate and answer five research questions....
Background and Objectives: Laryngeal cancer imposes a disproportionate burden on speech, airway protection and long-term quality of life. Contemporary population-based data for Central and Eastern Europe remain scarce, and the post-pandemic trajector...
Traditional intellectual property authentication relies on centralized intermediaries, which makes it difficult to address issues such as forgery, lack of trust, and opaque information. Combined with the characteristics of blockchain, such as decentr...