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Axioms
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Discrete Mathematics as the Basis and Application of Number Theory
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Discrete Mathematics as the Basis and Application of Number Theory

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 12418

Special Issue Editors

Prof. Dr. Takao Komatsu

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Guest Editor
Faculty of Education, Nagasaki University, Nagasaki 852-8521, Japan
Interests: combinatorics; discrete mathematics; number theory; pure mathematics; special functions
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Claudio Pita-Ruiz

E-Mail Website
Guest Editor
Faculty of Engineering, Universidad Panamericana, Mexico City Campus, Mexico
Interests: math; number theory; integer sequences and their properties; fibonacci sequences; fibonomials
Prof. Dr. Ram Krishna Pandey

E-Mail Website
Guest Editor
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
Interests: number theory (additive, combinatorial, and elementary)

Special Issue Information

Dear Colleagues, 

The subject of this Special Issue is discrete mathematics. At present, the use of AI is becoming indispensable in all situations and fields, and discrete mathematics is the basis of correct understanding for its use. Discrete mathematics covers a very wide range of fields, but this Special Issue focuses on its fundamental and applications in relevance to combinatorial number theory and elementary number theory. Any related or further idea is greatly welcomed if it does not contain only key words but also some interesting contents, such as sequences, sets, permutations, combinations, inductions, recurrence formulas, graphs, algorithms, and so on.

Prof. Dr. Takao Komatsu
Prof. Dr. Claudio Pita-Ruiz
Prof. Dr. Ram Krishna Pandey
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • discrete mathematics
  • generating functions
  • arithmetic functions
  • fibonacci numbers
  • bernoulli numbers
  • stirling numbers
  • harmonic numbers
  • matrices and determinants
  • continued fractions
  • recurrences
  • linear diophantine equations
  • additive bases
  • arithmetic progressions
  • numeration systems
  • beta-expansions
  • combinatorics on words

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (6 papers)

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Research

16 pages, 320 KiB  
Article
Series of Convergence Rate −1/4 Containing Harmonic Numbers
by Chunli Li and Wenchang Chu
Axioms 2023, 12(6), 513; https://doi.org/10.3390/axioms12060513 - 24 May 2023
Cited by 2 | Viewed by 991
Abstract
Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients. Among them, three conjectured identities due to Z.-W. Sun are also confirmed. Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
18 pages, 372 KiB  
Article
The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions
by Takao Komatsu and Claudio Pita-Ruiz
Axioms 2023, 12(2), 98; https://doi.org/10.3390/axioms12020098 - 17 Jan 2023
Cited by 9 | Viewed by 1574
Abstract
In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced [...] Read more.
In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p. Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
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7 pages, 225 KiB  
Article
Iterated Partial Sums of the k-Fibonacci Sequences
by Sergio Falcón and Ángel Plaza
Axioms 2022, 11(10), 542; https://doi.org/10.3390/axioms11100542 - 11 Oct 2022
Cited by 3 | Viewed by 2126
Abstract
In this paper, we find the sequence of partial sums of the k-Fibonacci sequence, say, Sk,n=∑j=1nFk,j, and then we find the sequence of partial sums of this new [...] Read more.
In this paper, we find the sequence of partial sums of the k-Fibonacci sequence, say, Sk,n=∑j=1nFk,j, and then we find the sequence of partial sums of this new sequence, Sk,n2)=∑j=1nSk,j, and so on. The iterated partial sums of k-Fibonacci numbers are given as a function of k-Fibonacci numbers, in powers of k, and in a recursive way. We finish the topic by indicating a formula to find the first terms of these sequences from the k-Fibonacci numbers themselves. Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
14 pages, 788 KiB  
Article
Cayley Graphs Defined by Systems of Equations
by Fuyuan Yang, Qiang Sun, Hongbo Zhou and Chao Zhang
Axioms 2022, 11(3), 100; https://doi.org/10.3390/axioms11030100 - 25 Feb 2022
Viewed by 2077
Abstract
Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley [...] Read more.
Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that BΓn is a Cayley graph with G=⟨ϕ,A⟩ a group of dihedral type if and only if the system Fn={fi|i=2,…,n} is Cayley graphic of dihedral type in R. As an application, the well-known Lova´sz Conjecture, which states that any finite connected Cayley graph has a Hamilton cycle, holds for the connected BΓn defined by Cayley graphic system Fn of dihedral type in the field GF(pk). Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
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18 pages, 293 KiB  
Article
Fourier Series for the Tangent Polynomials, Tangent–Bernoulli and Tangent–Genocchi Polynomials of Higher Order
by Cristina Bordaje Corcino and Roberto Bagsarsa Corcino
Axioms 2022, 11(3), 86; https://doi.org/10.3390/axioms11030086 - 22 Feb 2022
Cited by 3 | Viewed by 2089
Abstract
In this paper, the Fourier series expansion of Tangent polynomials of higher order is derived using the Cauchy residue theorem. Moreover, some variations of higher-order Tangent polynomials are defined by mixing the concept of Tangent polynomials with that of Bernoulli and Genocchi polynomials, [...] Read more.
In this paper, the Fourier series expansion of Tangent polynomials of higher order is derived using the Cauchy residue theorem. Moreover, some variations of higher-order Tangent polynomials are defined by mixing the concept of Tangent polynomials with that of Bernoulli and Genocchi polynomials, Tangent–Bernoulli and Tangent–Genocchi polynomials. Furthermore, Fourier series expansions of these variations are also derived using the Cauchy residue theorem. Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
9 pages, 265 KiB  
Article
Balances in the Set of Arithmetic Progressions
by Chan-Liang Chung, Chunmei Zhong and Kanglun Zhou
Axioms 2021, 10(4), 350; https://doi.org/10.3390/axioms10040350 - 20 Dec 2021
Viewed by 2333
Abstract
This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included. Full article
(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)
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