Ibrahim Sulaiman Ibrahim

The Tribonacci sequence, an extension of the Fibonacci sequence, exhibits rapid growth and a distinctive recursive structure. It plays a significant role in algebra, combinatorics, number theory, dynamical systems, and fractal geometry. Moreover, its applications extend to computer science — for modeling recursive algorithms, cryptographic structures — and to the natural sciences, such as population dynamics and biological growth patterns. In this study, we construct new sequence spaces based on the fusion of Tribonacci and lacunary sequences, generalized through modulus functions under appropriate conditions. We focus particularly on [ ]-Tribonacci lacunary statistical convergence of order [ ] and [ ]-strong Tribonacci lacunary summability of order [ ]. To strengthen our theoretical contributions, we integrate Neural Networks as computational tools for analyzing the convergence behavior and structural relationships among these sequence spaces. Several illustrative examples, supported by numerical simulations and figures, demonstrate the practical interpretation and effectiveness of the proposed concepts. Furthermore, refinements in lacunary sequences allow us to establish deeper interrelations, contributing to a more comprehensive understanding of these spaces and their applications in both mathematical theory and computational frameworks.

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.

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.

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.

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.

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.