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# Special Issue "Polynomials: Special Polynomials and Number-Theoretical Applications"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry".

Deadline for manuscript submissions: 31 July 2020.

## Special Issue Editor

Prof. Dr. Ákos Pintér
Website
Guest Editor
Institute of Mathematics, University of Debrecen, Debrecen, Hungary
Interests: polynomials; diophantine number theory

## Special Issue Information

Dear Colleagues,

The polynomials play an important role in mathematics and science. We aim to focus on two applications of these well-known mathematical objects in this Issue: special polynomials and number theory.
The special polynomials (including Bernoulli and Euler polynomials and their generalizations)  possess several applications in many branches of pure and applied mathematics. On the other hand, the nth Bernoulli polynomial Bn(X), for example, is a special bridge between certain mathematical topics; we refer here only to the classical formula by Jacob Bernoulli, 1k+2k+… +(x-1)k=1/(k+1)(Bk+1(x)-Bk+1(0))

The application of polynomials in number theory, especially in the theory of diophantine equations, goes back to the famous result of LeVeque from 1964. Let f(X) be a polynomial with rational coefficients, and let r1,…,rn denote the multiplicities of its zeros. LeVeque's theorem states that for given m>1 , the superelliptic equation  f(x) = ym in integers x, y has only finitely many solutions, unless {m/(m, r1),…, m/(m, rn)} is a permutation of one of the n-tuples {t, 1,..., 1}, t >0, and {2, 2, 1,..., 1}.
This was an ineffective finiteness result; later, several authors obtained effective versions providing an upper bound for x and y. However, one can see that to apply LeVeque's condition for a broad class of polynomial diophantine equations is a rather hard problem, because we have to determine the structure of zeros of an infinite family of polynomials.
The expected high-level articles should be a novel research contribution or an expository survey article related to the above-mentioned topics touching the role of symmetry.

Prof. Dr. Ákos Pintér
Guest Editor

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 papers will be 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. Symmetry 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 1400 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

• Polynomials
• Bernoulli- and Euler polynomials and their generalizations
• Other special polynomials
• Structure of zeros of polynomials
• Polynomial diophantine equations
• Superelliptic diophantine equations
• LeVeque’s theorem

## Published Papers (7 papers)

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# Research

Open AccessArticle
Durrmeyer-Type Generalization of Parametric Bernstein Operators
by Arun Kajla , Mohammad Mursaleen and Tuncer Acar
Symmetry 2020, 12(7), 1141; https://doi.org/10.3390/sym12071141 (registering DOI) - 08 Jul 2020
Abstract
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic [...] Read more.
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Open AccessFeature PaperArticle
Structure of Approximate Roots Based on Symmetric Properties of (p, q)-Cosine and (p, q)-Sine Bernoulli Polynomials
by Cheon Seoung Ryoo and Jung Yoog Kang
Symmetry 2020, 12(6), 885; https://doi.org/10.3390/sym12060885 - 30 May 2020
Abstract
This paper constructs and introduces $(p,q)$-cosine and $(p,q)$-sine Bernoulli polynomials using $(p,q)$-analogues of $(x+a)n$. Based on these polynomials, we discover basic properties [...] Read more.
This paper constructs and introduces $( p , q )$ -cosine and $( p , q )$ -sine Bernoulli polynomials using $( p , q )$ -analogues of $( x + a ) n$ . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using $( p , q )$ -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Show Figures Figure 1

Open AccessArticle
Symmetric Identities for Carlitz-Type Higher-Order Degenerate (p,q)-Euler Numbers and Polynomials
by Kyung-Won Hwang and Cheon Seoung Ryoo
Symmetry 2019, 11(12), 1432; https://doi.org/10.3390/sym11121432 - 20 Nov 2019
Abstract
The main goal of this paper is to investigate some interesting symmetric identities for Carlitz-type higher-order degenerate $(p,q)$-Euler numbers, and polynomials. At first, the Carlitz-type higher-order degenerate $(p,q)$-Euler numbers and polynomials are defined. [...] Read more.
The main goal of this paper is to investigate some interesting symmetric identities for Carlitz-type higher-order degenerate $( p , q )$ -Euler numbers, and polynomials. At first, the Carlitz-type higher-order degenerate $( p , q )$ -Euler numbers and polynomials are defined. We give few new symmetric identities for Carlitz-type higher-order degenerate $( p , q )$ -Euler numbers and polynomials. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Open AccessArticle
Certain Results for the Twice-Iterated 2D q-Appell Polynomials
by Hari M. Srivastava , Ghazala Yasmin , Abdulghani Muhyi and Serkan Araci
Symmetry 2019, 11(10), 1307; https://doi.org/10.3390/sym11101307 - 16 Oct 2019
Cited by 2
Abstract
In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated [...] Read more.
In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Show Figures Figure 1

Open AccessArticle
Some Symmetric Identities for Degenerate Carlitz-type (p, q)-Euler Numbers and Polynomials
by Kyung-Won Hwang and Cheon Seoung Ryoo
Symmetry 2019, 11(6), 830; https://doi.org/10.3390/sym11060830 - 24 Jun 2019
Cited by 2
Abstract
In this paper we define the degenerate Carlitz-type $(p,q)$-Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to [...] Read more.
In this paper we define the degenerate Carlitz-type $( p , q )$ -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type $( p , q )$ -Euler numbers and polynomials. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Open AccessArticle
On the Decomposability of the Linear Combinations of Euler Polynomials with Odd Degrees
by Ákos Pintér and Csaba Rakaczki
Symmetry 2019, 11(6), 739; https://doi.org/10.3390/sym11060739 - 31 May 2019
Abstract
In the present paper we prove that under certain conditions the linear combination of two Euler polynomials with odd degrees $Pn,m(x)=En(x)+cEm(x)$ is always indecomposable [...] Read more.
In the present paper we prove that under certain conditions the linear combination of two Euler polynomials with odd degrees $P n , m ( x ) = E n ( x ) + c E m ( x )$ is always indecomposable over $C$ , where c denotes a rational number. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
Open AccessArticle
Two Variables Shivley’s Matrix Polynomials
by Fuli He , Ahmed Bakhet , M. Hidan and M. Abdalla
Symmetry 2019, 11(2), 151; https://doi.org/10.3390/sym11020151 - 29 Jan 2019
Cited by 4
Abstract
The principal object of this paper is to introduce two variable Shivley’s matrix polynomials and derive their special properties. Generating matrix functions, matrix recurrence relations, summation formula and operational representations for these polynomials are deduced. Finally, Some special cases and consequences of our [...] Read more.
The principal object of this paper is to introduce two variable Shivley’s matrix polynomials and derive their special properties. Generating matrix functions, matrix recurrence relations, summation formula and operational representations for these polynomials are deduced. Finally, Some special cases and consequences of our main results are also considered. Full article
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)