Search Results (135)

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 converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(\pm 1)}^k}{w_{rk+l}{w}_{mk+s}{w}_{m(k+1)+s}}}$, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter m, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms. Full article

In this paper, we investigate the periodic behavior of incomplete Fibonacci and Lucas numbers modulo a positive integer m. Let $L(F,k,m)$ and $L(L,k,m)$ denote the least positive periods of the incomplete Fibonacci and incomplete Lucas sequences modulo m, respectively. We establish several fundamental properties of these periods, including a coprime-modulus lcm property that extends Wall’s classical result for the Fibonacci sequence. We derive explicit formulas for the least periods of both incomplete Fibonacci and incomplete Lucas sequences modulo arbitrary prime powers and arbitrary positive integers. Furthermore, we prove that the incomplete Fibonacci and incomplete Lucas sequences always have the same least period modulo m for every positive integer $m>1$ and every $k\ge 0$. Our approach is based on binomial coefficient representations, finite-difference operators, and periodicity properties of integer-valued polynomials. These results extend several classical periodicity properties of the Fibonacci and Lucas sequences to their incomplete counterparts and provide a systematic framework for studying their modular behavior. Full article

This paper presents an analytical investigation of a class of two-dimensional nonlinear discrete systems governed by bilinear difference equations. By leveraging the explicit representation theory for first-order bilinear recursions, we derive closed-form representations for the solution sequences under general assumptions on system parameters. Special attention is given to parameter configurations that lead to structural simplifications, such as symmetry-induced reductions to equivalent one-dimensional dynamics, thereby yielding more transparent analytical expressions. Several numerical examples are presented to illustrate the rich dynamical behavior of the proposed system, including oscillatory patterns and the interplay between its components. Full article

This paper introduces a sophisticated stochastic methodology grounded in a lattice rule featuring an optimized generating vector, which has been rigorously developed and comprehensively analyzed. At the heart of this investigation lies the Unified Danish Eulerian Model (UNI-DEM), an extensive large-scale mathematical framework designed to accurately capture the complex physical and chemical processes occurring within the atmosphere. The proposed lattice-based approach is systematically compared against state-of-the-art techniques, including the modified Sobol sequence and the Fibonacci lattice rule. The comparative analysis demonstrates the superiority of the proposed method in the estimation of high-dimensional integrals, highlighting its enhanced robustness and computational efficiency. These characteristics render it particularly well-suited for the computation of sensitivity indices, which are crucial for ensuring the reliability of scientific models. Furthermore, the study employs variance-based sensitivity analysis methods, notably the Sobol technique, to quantify the influence of input parameters on model outputs rigorously. A comprehensive experimental evaluation is undertaken, integrating advanced Monte Carlo algorithms in conjunction with stochastic scrambling strategies to further enhance computational performance. In addition, the research examines the effects of varying emission levels on key atmospheric pollutants such as ammonia, ozone, ammonium sulfate, and ammonium nitrate, with particular emphasis on major European urban centers exhibiting diverse geographical and environmental conditions. These results underscore the critical importance of sensitivity analysis in validating model accuracy and elucidating the intricate relationships between input parameters and environmental outcomes. Full article

Can you stretch and reform a curve such that it fills a square completely? This question dates back to the late 19th century—the origin of space-filling curves. It was proven affirmatively by many great mathematicians. In this document, we reconsider the problem and present a different proof using the Cartesian product of Fibonacci substitution with itself. Our construction differs from other curves by the particular ordering produced by the Cartesian product of Fibonacci substitutions and the iterative geometric design induced by it. Full article

This work presents an exact microcanonical combinatorial analysis of the one-dimensional antiferromagnetic Ising model. At the primary ground-state level crossing $B/J=2$, degeneracies follow the Fibonacci and Lucas sequences for open chains and periodic rings, respectively. We extend this framework to the complete excitation spectrum, demonstrating that the density of states is constructed from topological defects governed by linear Diophantine equations and p-fold Fibonacci convolutions. Open boundaries act as fractional defects, densifying the chain spectrum into energy steps of $2J$, whereas the closed ring remains quantized in units of $4J$. Notably, this exact topological counting exposes non-trivial spectral gaps near the fully polarized limit, strictly forbidding the penultimate macroscopic energy levels in both topologies. Using the transfer-matrix formalism, we derive exact closed-form expressions for the critical degeneracies at all energy levels. These results provide a rigorous analytical foundation for extracting exact residual entropies and exposing the intrinsic number-theoretic architecture of quantum critical manifolds. Full article

In this study, we propose a matrix-based transformation framework constructed from special integer sequences, including Fibonacci, Lucas, Pell, and Jacobsthal numbers. The approach is based on block-wise $2\times 2$ matrix transformations that preserve key structural invariants, particularly the determinant, ensuring explicit invertibility of the scheme. By combining multiple recurrence-based matrices within a unified framework, the method provides flexible forward and inverse transformations without increasing matrix dimensions or introducing additional redundancy. The determinant-preserving property enables intrinsic consistency checking and supports an analytic error-detection and correction mechanism at the block level. Several illustrative examples are presented to demonstrate the applicability of the proposed scheme and its computational characteristics. The framework is purely algebraic and can be extended to other matrix families generated by linear recurrence relations, making it suitable for a wide range of applications in applied and computational mathematics. Full article

Polylogarithm-weighted sequences and $h\left(x\right)$-Fibonacci/Lucas polynomials have each been studied extensively, but a common formulation that incorporates generalized hyperharmonic weights into both these kernels and related Legendre-type kernels has not been formulated in a unified way. In this paper, the classical generating functions are deformed by the factor ${\mathrm{Li}}_p\left(t\right)/{(1-t)}^q$, and the resulting coefficients are derived by Cauchy product arguments. This construction yields the $h\left(x\right)$-Fibonacci–polylogarithm and $h\left(x\right)$-Lucas–polylogarithm polynomials, explicit coefficient formulas, convolution identities, recurrence relations, and parity properties, together with a unified two-parameter family of generalized $h\left(x\right)$-Fibonacci–Lucas–polylogarithm polynomials ${\mathbb{P}}_{h,n}^{a,b,p,q}\left(x\right)$. The same deformation principle also gives rise to Legendre–polylogarithm polynomials and to a $(q,\lambda )$-extension obtained from a weighted Legendre generating kernel. These families provide a natural generating-function setting for models in which cumulative harmonic or hyperharmonic effects are intrinsic, while also making explicit the main analytic restrictions of the deformation, including convergence constraints and the loss of classical orthogonality in the Legendre setting. Full article

Simple recursive rules often conceal surprisingly complex structures, and the Tribonacci sequence is an interesting example of how elementary arithmetic can lead to complex and visually expressive patterns. In this research paper, we present triangular arrays generated by a Petri net encoding of the Tribonacci recurrence, extending our previous Fibonacci construction. Token propagation realizes the delayed dependencies of the recurrence and gives rise to a directed global triangular geometry. Parity and modular colourings highlight persistent self-similar motifs reminiscent of classical fractal patterns. We also outline higher-order k-bonacci extensions and compare the resulting visual structures. Within this framework, global fractal-like visibility tends to degrade as the order increases, while extreme orders can enter a different limiting regime. Full article

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ${\sum }_{n\ge 0}{\left|a_n\right|}^2/F_{n+1}$, we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ${(1-t-t^2)}^{-1}$ and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ${\rho }_F={\phi }^{-1/2}$ (the square-root inverse of the golden ratio $\phi$). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of $\mathbb{BC}$, compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat $(p,q)$-Fibonacci weights, obtaining a one-parameter family of rational kernels ${(1-pt-q{t}^2)}^{-1}$ and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts. Full article

We propose a design of a composite splitter–filter by replacing the traditional periodic arrays with Fibonacci rod chains along both sides of the output channel of a T-junction photonic crystal waveguide. This integrated structure concurrently realizes the dual functions of a power splitter and an optical filter. The coexistence and effectiveness of these two functions are verified through numerical simulations. Furthermore, the proposed device exhibits excellent robustness against three types of defects and enables strong tunability of its operating wavelength window. Owing to these superior characteristics, this hybrid photonic crystal–quasicrystal structure holds significant application potential in photonic integrated circuits and high-performance optical communication systems. Full article

Mathematics, as Bertrand Russell noted, possesses both truth and beauty. In this work, we revisit the classical Fibonacci recurrence thanks to a minimal Petri net. Starting from a minimal layered construction that mirrors the second-order additive rule $F_n=F_{n-1}+F_{n-2}$, we show that the marking dynamics of the associated net generate a combinatorial triangle whose parity structure reveals a self-similar, Sierpiński-like pattern. To the best of our knowledge, this oblique fractal geometry has never been formally documented. We provide a formal definition of the underlying Petri net, analyse its computational properties, and explore the emergence of higher-order harmonics when token markings are considered modulo primes. The study highlights how a classical recurrence gives rise to previously unnoticed geometric regularities at the intersection of mathematics and art. Beyond its mathematical interest, the construction illustrates how minimal Petri net dynamics can be used as formal specification patterns for distributed, event-driven systems. Full article

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

We investigate the multifractal geometry of irregular sets arising from weighted averages of random variables, where the weights $\left(w_n\right)$ form a positive sequence with exponential growth. Our analysis applies in particular to sequences generated by linear recurrence relations of Fibonacci type, including higher-order generalizations such as the Tetranacci sequence $\left({\mathsf{T}}_n\right)$. Using a Cantor-type construction built from alternating free and forced blocks, we show that the associated exceptional sets may attain full Hausdorff and packing dimension, independently of the precise form of the recurrence. We further develop a probabilistic interpretation of $\left({\mathsf{T}}_n\right)$ through an appropriate Markov representation that encodes its combinatorial evolution and yields sharp asymptotic behavior. Finally, given $n+1$ consecutive terms of a Fibonacci-type sequence, one may construct a polynomial $P_n\left(x\right)$ of degree at most n via Lagrange interpolation; we show that this polynomial admits an implicit recursive representation consistent with the underlying recurrence. Full article

The limited and often irreplaceable battery energy of Wireless Sensor Network (WSN) nodes, which are typically deployed in harsh environments, poses a critical challenge. Excessive energy consumption can lead to node failure and consequent data loss, making energy efficiency a central research focus. To address the limitations of the LEACH protocol in cluster head (CH) election and transmission modes, this paper proposes an optimized approach. First, sensor nodes are clustered using a Self-Organizing Map (SOM) neural network. Subsequently, the CH election function incorporates the node’s residual energy, distance to the base station, and neighbor node density. Finally, the data transmission stage employs a hybrid method combining Fibonacci sequences and a bee algorithm for routing optimization. The simulation results demonstrate that the proposed protocol outperforms benchmarks in terms of the node death round, network lifetime, and data throughput across different base station locations, offering a valuable technical solution for routing optimization in medium- and large-scale WSNs. Full article