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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 30 September 2025 | Viewed by 2600

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Special Issue Editors

Prof. Dr. Francisco Javier Pérez Fernández

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Guest Editor

Department of Mathematics, Faculty of Sciences, University of Cádiz, 11510 Puerto Real, Spain
Interests: functional analysis; mostly in the filed of series and summability in normed spaces; mathematical foundations from the perspective of the history of mathematical analysis

Dr. María C. Listán-García

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Department of Mathematics, Universidad de Cádiz, Apdo. 40, 11510 Puerto Real, Spain
Interests: functional analysis
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Prof. Dr. Soledad Moreno-Pulido

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Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain
Interests: functional analysis (sumability, measure theory, effect algebras, norm attaining operators…) and applied mathematics lines (multifractal analysis of complex networks; optimisation with supporting vectors, databases and statistical analysis in numismatics

Special Issue Information

Dear Colleagues,

This Special Issue focuses on novel and original results related to summability and convergence methods such as Cesàro summability, statistical convergence, lacunary summability and matrix methods. Papers that address recent developments in Boolean and effect algebras are also welcome.

The aim of this Special Issue is to present recent developments in summability and convergence methods, including, but not limited to, the following:

Multiplier Convergent Series
Statistical convergence
Cesáro summabiliy and convergence
Lacunary convergence
Boolean and Effect algebras
Abelian theorems
Tauberian theorems

The framework employed typically encompasses Banach spaces but can be as general as topological groups or specialization in finite dimensional normed spaces.

Prof. Dr. Francisco Javier Pérez Fernández
Dr. María C. Listán-García
Prof. Dr. Soledad Moreno-Pulido
Guest Editors

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Keywords

series
asymptotically statistical equivalence
difference sequences
statistical convergence
multiplier convergent series
matrix methods
modulus function
statistical convergence in probability
Boolean algebras
effect algebras
Cesàro summability
lacunary convergence
f-statistical convergence

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Published Papers (3 papers)

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Research

14 pages, 286 KiB

Open AccessArticle

Deferred f-Statistical Convergence of Generalized Difference Sequences of Order α

by Fernando León-Saavedra, Mikail Et and Fatih Temizsu

Mathematics 2025, 13(3), 463; https://doi.org/10.3390/math13030463 - 30 Jan 2025

Viewed by 315

Abstract

Studies on difference sequences was introduced in the 1980s, and since then, many mathematicians have studied this kind of sequences and obtained some generalized difference sequence spaces. In this paper, using the generalized difference operator, we introduce the concept of the deferred f [...] Read more.

α

and give some inclusion relations between the deferred f-statistical convergence of generalized difference sequences and deferred f-statistical convergence of generalized difference sequences of the order

α

. Our results are more general than the corresponding results in the existing literature. Full article

(This article belongs to the Special Issue Summability and Convergence Methods)

43 pages, 619 KiB

Open AccessArticle

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

by Leonid G. Fel

Mathematics 2025, 13(2), 281; https://doi.org/10.3390/math13020281 - 16 Jan 2025

Viewed by 402

Abstract

We consider a wide class of summatory functions

F \{f; N, p^{m}\} = \sum_{k \leq N} f (p^{m} k)

m \in Z_{+} \cup {0}

associated with the multiplicative arithmetic functions f of a scaled [...] Read more.

We consider a wide class of summatory functions

F \{f; N, p^{m}\} = \sum_{k \leq N} f (p^{m} k)

m \in Z_{+} \cup {0}

associated with the multiplicative arithmetic functions f of a scaled variable

k \in Z_{+}

, where p is a prime number. Assuming an asymptotic behavior of the summatory function,

F {f; N, 1} \overset{N \to \infty}{=} G_{1} (N) [1 + O (G_{2} (N))]

, where

G_{1} (N) = N^{a_{1}} {(\log N)}^{b_{1}}

G_{2} (N) = N^{- a_{2}} {(\log N)}^{- b_{2}}

and

a_{1}, a_{2} \geq 0

- \infty < b_{1}, b_{2} < \infty

, we calculate the renormalization function

R (f; N, p^{m})

, defined as a ratio

F \{f; N, p^{m}\} / F {f; N, 1}

, and find its asymptotics

R_{\infty} (f; p^{m})

when

N \to \infty

. We prove that a renormalization function is multiplicative, i.e.,

R_{\infty} (f; \prod_{i = 1}^{n} p_{i}^{m_{i}}) = \prod_{i = 1}^{n} R_{\infty} (f; p_{i}^{m_{i}})

with n distinct primes

p_{i}

. We extend these results to the other summatory functions

\sum_{k \leq N} f (p^{m} k^{l})

m, l, k \in Z_{+}

and

\sum_{k \leq N} \prod_{i = 1}^{n} f_{i} (k p^{m_{i}})

f_{i} \neq f_{j}

m_{i} \neq m_{j}

. We apply the derived formulas to a large number of basic summatory functions including the Euler

ϕ (k)

and Dedekind

ψ (k)

totient functions, divisor

σ_{n} (k)

and prime divisor

β (k)

functions, the Ramanujan sum

C_{q} (n)

and Ramanujan

τ

Dirichlet series, and others. Full article

(This article belongs to the Special Issue Summability and Convergence Methods)

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Figure 1

13 pages, 267 KiB

Open AccessArticle

A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions

by Ibrahim S. Ibrahim and María C. Listán-García

Mathematics 2024, 12(17), 2718; https://doi.org/10.3390/math12172718 - 30 Aug 2024

Cited by 1 | Viewed by 1044

Abstract

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. Full article

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