Special Issue "Current Trends in Symmetric Polynomials with Their Applications Ⅱ"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 30 November 2019.

Special Issue Editor

Prof. Taekyun Kim
E-Mail Website
Guest Editor
Department of Mathematics, College of Natural Science, Kwangwoon University, Seoul 139-704, S. Korea
Tel. +82-(0)2-940-8368
Interests: integral equations
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

The symmetric polynomials have various applications in many branches of mathematics and mathematical physics. These polynomials are defined by linear polynomials (differential relations), globally referred to as functional equations that arise in well-defined combinatorial contexts, and they lead systematically to well-defined classes of functions. The symmetric functions for the sequence of polynomials are used in analyzing sequences of functions, in finding a closed formula for a sequence, in finding recurrence relations and differential equations, in relationships between sequences, in asymptotic behavior of sequences, and in proving identities involving sequences.

We aim to design this special issue for researchers with an interest in pure and applied Mathematics. This special issue aims to present theory, methods, and applications of recent/current symmetric polynomials.

Each paper that will be published in this special issue aims at enriching the understanding of current research problems, theories, and applications on the chosen topics. The emphasis will be to present the basic developments concerning an idea in full detail, and also contain the most recent advances made in the area of symmetric functions and polynomials.

Advanced research on symmetric functions and polynomials is essential to study and model various changes in their natures. We will attempt to include some carefully selected papers in these areas of research that have significant applications. Much applicable mathematics cannot be investigated further or used without the applications of symmetric special functions and polynomials.

Thus, this special issue is expected to be beneficial for researchers who are interested in mathematics that has applications in pure and applied mathematics and uses tools mainly from the broad mathematical grouping.

Prof. Taekyun Kim
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

  • Symmetric polynomials
  • Special Functions
  • Special polynomials
  • Inequalities
  • Integral Equations
  • Mathematical Physics
  • Bosonic p-adic integral
  • Fermionic p-adic integral

Published Papers (11 papers)

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Research

Open AccessArticle
A New Identity Involving Balancing Polynomials and Balancing Numbers
Symmetry 2019, 11(9), 1141; https://doi.org/10.3390/sym11091141 - 07 Sep 2019
Abstract
In this paper, a second-order nonlinear recursive sequence M ( h , i ) is studied. By using this sequence, the properties of the power series, and the combinatorial methods, some interesting symmetry identities of the structural properties of balancing numbers and balancing [...] Read more.
In this paper, a second-order nonlinear recursive sequence M ( h , i ) is studied. By using this sequence, the properties of the power series, and the combinatorial methods, some interesting symmetry identities of the structural properties of balancing numbers and balancing polynomials are deduced. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
Note on Type 2 Degenerate q-Bernoulli Polynomials
Symmetry 2019, 11(7), 914; https://doi.org/10.3390/sym11070914 - 13 Jul 2019
Cited by 3
Abstract
The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, [...] Read more.
The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials
Symmetry 2019, 11(7), 847; https://doi.org/10.3390/sym11070847 - 01 Jul 2019
Cited by 3
Abstract
In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer [...] Read more.
In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence
Symmetry 2019, 11(6), 788; https://doi.org/10.3390/sym11060788 - 13 Jun 2019
Cited by 1
Abstract
The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of [...] Read more.
The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
On r-Central Incomplete and Complete Bell Polynomials
Symmetry 2019, 11(5), 724; https://doi.org/10.3390/sym11050724 - 27 May 2019
Abstract
Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions [...] Read more.
Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
Some Identities of Fully Degenerate Bernoulli Polynomials Associated with Degenerate Bernstein Polynomials
Symmetry 2019, 11(5), 709; https://doi.org/10.3390/sym11050709 - 24 May 2019
Abstract
In this paper, we investigate some properties and identities for fully degenerate Bernoulli polynomials in connection with degenerate Bernstein polynomials by means of bosonic p-adic integrals on Z p and generating functions. Furthermore, we study two variable degenerate Bernstein polynomials and the [...] Read more.
In this paper, we investigate some properties and identities for fully degenerate Bernoulli polynomials in connection with degenerate Bernstein polynomials by means of bosonic p-adic integrals on Z p and generating functions. Furthermore, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
The Power Sums Involving Fibonacci Polynomials and Their Applications
Symmetry 2019, 11(5), 635; https://doi.org/10.3390/sym11050635 - 06 May 2019
Abstract
The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized [...] Read more.
The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessFeature PaperArticle
Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials
Symmetry 2019, 11(5), 613; https://doi.org/10.3390/sym11050613 - 02 May 2019
Cited by 6
Abstract
The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime [...] Read more.
The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
Extended Degenerate r-Central Factorial Numbers of the Second Kind and Extended Degenerate r-Central Bell Polynomials
Symmetry 2019, 11(4), 595; https://doi.org/10.3390/sym11040595 - 24 Apr 2019
Cited by 1
Abstract
In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell [...] Read more.
In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell polynomials, and also degenerate versions of the extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, all of which have been studied by Kim and Kim. We study various properties and identities concerning those numbers and polynomials and also their connections. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
The Extended Minimax Disparity RIM Quantifier Problem
Symmetry 2019, 11(4), 481; https://doi.org/10.3390/sym11040481 - 03 Apr 2019
Abstract
An interesting regular increasing monotone (RIM) quantifier problem is investigated. Amin and Emrouznejad [Computers & Industrial Engineering 50(2006) 312–316] have introduced the extended minimax disparity OWA operator problem to determine the OWA operator weights. In this paper, we propose a corresponding continuous extension [...] Read more.
An interesting regular increasing monotone (RIM) quantifier problem is investigated. Amin and Emrouznejad [Computers & Industrial Engineering 50(2006) 312–316] have introduced the extended minimax disparity OWA operator problem to determine the OWA operator weights. In this paper, we propose a corresponding continuous extension of an extended minimax disparity OWA model, which is the extended minimax disparity RIM quantifier problem, under the given orness level and prove it analytically. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
Open AccessArticle
The Solution Equivalence to General Models for the RIM Quantifier Problem
Symmetry 2019, 11(4), 455; https://doi.org/10.3390/sym11040455 - 01 Apr 2019
Abstract
Hong investigated the relationship between the minimax disparity minimum variance regular increasing monotone (RIM) quantifier problems. He also proved the equivalence of their solutions to minimum variance and minimax disparity RIM quantifier problems. Hong investigated the relationship between the minimax ratio and maximum [...] Read more.
Hong investigated the relationship between the minimax disparity minimum variance regular increasing monotone (RIM) quantifier problems. He also proved the equivalence of their solutions to minimum variance and minimax disparity RIM quantifier problems. Hong investigated the relationship between the minimax ratio and maximum entropy RIM quantifier problems and proved the equivalence of their solutions to the maximum entropy and minimax ratio RIM quantifier problems. Liu proposed a general RIM quantifier determination model and proved it analytically by using the optimal control technique. He also gave the equivalence of solutions to the minimax problem for the RIM quantifier. Recently, Hong proposed a modified model for the general minimax RIM quantifier problem and provided correct formulation of the result of Liu. Thus, we examine the general minimum model for the RIM quantifier problem when the generating functions are Lebesgue integrable under the more general assumption of the RIM quantifier operator. We also provide a solution equivalent relationship between the general maximum model and the general minimax model for RIM quantifier problems, which is the corrected and generalized version of the equivalence of solutions to the general maximum model and the general minimax model for RIM quantifier problems of Liu’s result. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
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