2024-11

PLoS ONE (Issue : 11) (Volume : 19)

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.

2024-08

Journal of Applied Analysis

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.

2024-08

Mathematics (Issue : 17) (Volume : 12)

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.

2024-06

Kyungpook Mathematical Journal (Issue : 2) (Volume : 64)

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.

2024-05

Boletim da Sociedade Paranaense de Matemática (Volume : 42)

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

2024-02

Rendiconti del Circolo Matematico di Palermo

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.

2021-08

Annals of the University of Craiova - Mathematics and Computer Science Series (Issue : 1) (Volume : 48)

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.