In this paper we propose a new perspective related to Lucas sequence with statistical and summability, some considerations have been given in connection with their applications of signal processing. We define the notions of $lambda$-Lucas statistically convergent and strongly $lambda$-Lucas summability through modulus functions. The study of investigates on the Lucas transform and its embedding diagram into the representative form of Lucas numbers is studied in the context of convergence and summability problems. Furthermore, we provide inclusions as well as principles of equivalence, which lead to conditions guaranteeing the uniqueness of restrictions and generalize statistical convergence theory. Numerical simulations on a wide variety of signals such as noisy sinusoidal signals and blurred images demonstrate the flexibility of the proposed approach. These experiments show a steady decay to zero, which implies considerable noise immunity. In addition, by providing alternative perspectives of assessing signal quality, compression strategies and filtering techniques, those methods help in distinguishing the relevant signal information from background noise. Bringing together theoretical results and practical real-world signal analysis examples, the paper introduces classical and recent contributions to summability theory as well as new signal-processing procedures.

The aim of this study is to define $\lambda_f$-density, $\lambda_f$-statistical convergence and $\lambda_f$-statistical boundedness in metric spaces by using modulus functions under different conditions. In addition, we obtain some relations between the sets of $\lambda_f$-statistically convergent and $\lambda_f$-statistically bounded sequences in metric spaces.

The aim of this study is to define $\lambda_f$-density, $\lambda_f$-statistical convergence and $\lambda_f$-statistical boundedness in metric spaces by using modulus functions under different conditions. In addition, we obtain some relations between the sets of $\lambda_f$-statistically convergent and $\lambda_f$-statistically bounded sequences in metric spaces.

In this paper, we extend the concepts of statistical convergence and strong summability for the sequences of fuzzy numbers using modulus functions. By introducing appropriate conditions on the modulus functions, we generalize and refine existing notions of convergence within the fuzzy setting. Additionally, we establish several interrelationships between these extended concepts, thereby contributing to the deeper understanding of summability and convergence behavior in the sequences of fuzzy numbers.

The study of sequence convergence and summability has made significant strides, yet exploring alternative frameworks remains a promising avenue of research. This paper introduces a novel approach to the convergence and boundedness of triple sequences, utilizing Bessel functions in combination with modulus functions. This framework provides a refined analytical perspective on sequence behavior in higher dimensions. We construct new sequence spaces based on strong Bessel summability and investigate their structural properties, such as inclusion relations, completeness, and other key characteristics. Additionally, we incorporate Artificial Neural Networks (ANNs) for the classification of sequences generated by linear, quadratic, and exponential functions. Through the integration of ANN models, we demonstrate their ability to classify these sequences with high accuracy, showcasing the potential of machine learning techniques in advancing sequence space theory. Several illustrative examples are provided to highlight the effectiveness of both the theoretical and computational frameworks. The results of this study contribute to the ongoing development of sequence space theory, with implications for future research in multidimensional summability, convergence analysis, and machine learning applications in sequence classification.

This paper explores new directions in the study of sequence spaces for fuzzy numbers by introducing and examining Bessel statistical convergence and strong Bessel summability. We establish key inclusion relations and build a theoretical framework that deepens the understanding of convergence and summability in fuzzy settings. As an important application, we extend the classical fuzzy Korovkin-type theorem through Bessel statistical convergence, showing greater flexibility and wider applicability than existing results. Our framework is particularly effective for sequences that fail to converge in the classical sense but display statistical convergence, as illustrated with Bessel-perturbed approximation operators. In addition, we investigate the fuzzy rate of Bessel statistical convergence, providing a way to measure the speed of convergence and offering new insight into approximation processes in fuzzy contexts. Examples and visualizations are included to demonstrate the practical value of our results in areas such as uncertainty modeling, control systems, and computational mathematics.

The statistical convergence with respect to a modulus function has various applications in both mathematics and statistics. The main purpose of this research paper is to establish the relations between the sets of strongly f-lacunary summable and strongly g-lacunary summable sequences, strongly f-lacunary summable and g-lacunary statistically convergent sequences, where f and g are different modulus functions under certain conditions. Furthermore, for some special modulus functions, we establish the relations between the sets of strongly f-lacunary summable and strongly lacunary summable sequences.

Correction: Bessel statistical convergence: New concepts and applications in sequence theory Page 1 PLOS One | https://doi.org/10.1371/journal.pone.0329900 August 5, 2025 1 / 1 OPEN ACCESS Citation: Ibrahim IS, Yousif MA, Mohammed PO, Baleanu D, Zeeshan A, Abdelwahed M (2025) Correction: Bessel statistical convergence: New concepts and applications in sequence theory. PLoS One 20(8): e0329900. https://doi. org/10.1371/journal.pone.0329900 Published: August 5, 2025

This paper explores the dynamic behavior of optical soliton solutions for the modied Kawahara (mK) equation and the modied Benjamin- Bona-Mahony (mBBM) equation, two signicant nonlinear evolution equations. Using an advanced analytical approach, a diverse set of soliton solutions is derived, including bell-shaped, anti-bell-shaped, W-shaped, M-shaped, and periodic waveforms. ese solutions unveil the intricate nonlinear dynamics underlying the equations. e robustness of the method is demonstrated through comprehensive 2D, 3D, and contour visualizations, ošering clear insights into the physical signicance of the solitons. e study enhances the existing catalog of soliton solutions, contributing to a deeper understanding of nonlinear wave propagation and its potential applications in elds such as optical communication and žuid dynamics.

A Fibonacci sequence has profound applications in various fields, including mathematics, computer science, and nature. From modeling natural phenomena to optimizing algorithms and cryptography, Fibonacci sequences play a pivotal role. Moreover, their connection to the golden ratio underscores their relevance in art, architecture, and aesthetics. In this research paper, we introduce new versions of convergence and summability of sequences of numbers with the help of the Fibonacci sequence called λ-Fibonacci statistical convergence and strong λ-Fibonacci summability of sequences of numbers by using modulus functions. Also, the definition of -Fibonacci statistical Cauchy of sequences of numbers is given by using modulus functions. These concepts are supported by several of significant theorems and properties in the study. Furthermore, some relations related to these concepts are obtained.

This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.

The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.

This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines.

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators.

zoom_out_map search menu Journals Mathematics Volume 12 Issue 23 10.3390/math12233764 Download PDFsettingsOrder Article Reprints This is an early access version, the complete PDF, HTML, and XML versions will be available soon. Open AccessArticle Weighted Statistical Convergence and Cluster Points: The Fibonacci Sequence-Based Approach Using Modulus Functions by Ibrahim S. Ibrahim 1ORCID,Iver Brevik 2ORCID,Ravi P. Agarwal 3ORCID,Majeed A. Yousif 1ORCID,Nejmeddine Chorfi 4ORCID andPshtiwan Othman Mohammed 5,*ORCID 1 Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq 2 Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 3 Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA 4 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 5 Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq * Author to whom correspondence should be addressed. Mathematics 2024, 12(23), 3764; https://doi.org/10.3390/math12233764 Submission received: 1 November 2024 / Revised: 20 November 2024 / Accepted: 27 November 2024 / Published: 28 November 2024 (This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions, 3rd Edition) Downloadkeyboard_arrow_down Versions Notes Abstract In this paper, the Fibonacci sequence, renowned for its significance across various fields, its ability to illuminate numerical concepts, and its role in uncovering patterns in mathematics and nature, forms the foundation of this research. This study introduces innovative concepts of weighted density, weighted statistical summability, weighted statistical convergence, and weighted statistical Cauchy, uniquely defined via the Fibonacci sequence and modulus functions. Key theorems, relationships, examples, and properties substantiate these novel principles, advancing our comprehension of sequence behavior. Additionally, we extend the notion of statistical cluster points within a broader framework, surpassing traditional definitions and offering deeper insights into convergence in a statistical context. Our findings in this paper open avenues for new applications and further exploration in various mathematical fields.

In this paper, we explore multi positive solutions together with their existence and uniqueness, which is properly defined for delta fractional version of Riemann–Liouville difference operators. Our exploration encompasses two distinct directions. In the first direction, we construct the Green’s function formula for the proposed delta fractional boundary value problems of order , and we present some essential properties of this function. The last and main results suggest using the well-known fixed point theorems in a Banach space for testing the existing and uniqueness of multi-positive solutions of such problems.

The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces.

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 known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

In this paper, we establish relations between the sets of strongly Cesàro summable sequences of complex numbers for modulus functions f and g satisfying various conditions. Furthermore, for some special modulus functions, we obtain relations between the sets of strongly Cesàro summable and statistically convergent sequences of complex numbers.

In this research paper, we introduce some concepts of λf−density in connection with modulus functions under certain conditions. Furthermore, we establish some relations between the sets of λf−statistically convergent and λf−statistically bounded sequences

In this research paper, our aim is to introduce the concepts of (α,β)-statistical convergence of order γ and (α,β)-statistical boundedness of order γ by using unbounded modulus functions in metric spaces. Furthermore, we obtain some relations between these concepts, and we provide some counterexamples to support our results.

This research paper focuses on defining the relationships between the sets of strongly lacunary summable and lacunary statistically convergent sequences by using different modulus functions f and g under certain conditions and different orders. Furthermore, for some special modulus functions, we establish the relations between the sets of strongly f-lacunary summable sequences and strongly f-lacunary summable sequences of order alpha.