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Symmetry
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Current Trends in Symmetric Polynomials with Their Applications Ⅱ
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Current Trends in Symmetric Polynomials with Their Applications Ⅱ

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

Deadline for manuscript submissions: closed (31 March 2020) | Viewed by 34064

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Taekyun Kim

E-Mail Website
Guest Editor
Department of Mathematics, College of Natural Science, Kwangwoon University, Seoul 139-704, Republic of Korea
Interests: number theory; analysis and applications; stochastic process
Special Issues, Collections and Topics 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 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. 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 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

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

Published Papers (18 papers)

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10 pages, 209 KiB  
Article
Some Identities on the Poly-Genocchi Polynomials and Numbers
by Dmitry V. Dolgy and Lee-Chae Jang
Symmetry 2020, 12(6), 1007; https://doi.org/10.3390/sym12061007 - 14 Jun 2020
Cited by 9 | Viewed by 1793
Abstract
Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive [...] Read more.
Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
8 pages, 729 KiB  
Article
An Erdős-Ko-Rado Type Theorem via the Polynomial Method
by Kyung-Won Hwang, Younjin Kim and Naeem N. Sheikh
Symmetry 2020, 12(4), 640; https://doi.org/10.3390/sym12040640 - 17 Apr 2020
Cited by 1 | Viewed by 2049
Abstract
A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [...] Read more.
A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
16 pages, 270 KiB  
Article
A Note on Parametric Kinds of the Degenerate Poly-Bernoulli and Poly-Genocchi Polynomials
by Taekyun Kim, Waseem A. Khan, Sunil Kumar Sharma and Mohd Ghayasuddin
Symmetry 2020, 12(4), 614; https://doi.org/10.3390/sym12040614 - 13 Apr 2020
Cited by 12 | Viewed by 1650 | Correction
Abstract
Recently, the parametric kind of some well known polynomials have been presented by many authors. In a sequel of such type of works, in this paper, we introduce the two parametric kinds of degenerate poly-Bernoulli and poly-Genocchi polynomials. Some analytical properties of these [...] Read more.
Recently, the parametric kind of some well known polynomials have been presented by many authors. In a sequel of such type of works, in this paper, we introduce the two parametric kinds of degenerate poly-Bernoulli and poly-Genocchi polynomials. Some analytical properties of these parametric polynomials are also derived in a systematic manner. 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)
9 pages, 248 KiB  
Article
Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind
by Taekyun Kim, Lee-Chae Jang, Dae San Kim and Han Young Kim
Symmetry 2020, 12(4), 510; https://doi.org/10.3390/sym12040510 - 02 Apr 2020
Cited by 23 | Viewed by 2044
Abstract
In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study [...] Read more.
In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
14 pages, 321 KiB  
Article
Exceptional Set for Sums of Symmetric Mixed Powers of Primes
by Jinjiang Li, Chao Liu, Zhuo Zhang and Min Zhang
Symmetry 2020, 12(3), 367; https://doi.org/10.3390/sym12030367 - 02 Mar 2020
Viewed by 1769
Abstract
The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of [...] Read more.
The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
7 pages, 216 KiB  
Article
Some Identities and Inequalities Involving Symmetry Sums of Legendre Polynomials
by Tingting Wang and Liang Qiao
Symmetry 2019, 11(12), 1521; https://doi.org/10.3390/sym11121521 - 16 Dec 2019
Cited by 2 | Viewed by 1684
Abstract
By using the analysis methods and the properties of Chebyshev polynomials of the first kind, this paper studies certain symmetry sums of the Legendre polynomials, and gives some new and interesting identities and inequalities for them, thus improving certain existing results. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
7 pages, 228 KiB  
Article
A New Identity Involving Balancing Polynomials and Balancing Numbers
by Yuanyuan Meng
Symmetry 2019, 11(9), 1141; https://doi.org/10.3390/sym11091141 - 07 Sep 2019
Cited by 5 | Viewed by 1783
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)
10 pages, 255 KiB  
Article
Note on Type 2 Degenerate q-Bernoulli Polynomials
by Dae San Kim, Dmitry V. Dolgy, Jongkyum Kwon and Taekyun Kim
Symmetry 2019, 11(7), 914; https://doi.org/10.3390/sym11070914 - 13 Jul 2019
Cited by 3 | Viewed by 1805
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)
13 pages, 244 KiB  
Article
Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials
by Dmitry V. Dolgy, Dae San Kim, Jongkyum Kwon and Taekyun Kim
Symmetry 2019, 11(7), 847; https://doi.org/10.3390/sym11070847 - 01 Jul 2019
Cited by 5 | Viewed by 2015
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)
10 pages, 247 KiB  
Article
Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence
by Zhuoyu Chen and Lan Qi
Symmetry 2019, 11(6), 788; https://doi.org/10.3390/sym11060788 - 13 Jun 2019
Cited by 10 | Viewed by 1742
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)
12 pages, 267 KiB  
Article
On r-Central Incomplete and Complete Bell Polynomials
by Dae San Kim, Han Young Kim, Dojin Kim and Taekyun Kim
Symmetry 2019, 11(5), 724; https://doi.org/10.3390/sym11050724 - 27 May 2019
Cited by 3 | Viewed by 2187
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)
11 pages, 231 KiB  
Article
Some Identities of Fully Degenerate Bernoulli Polynomials Associated with Degenerate Bernstein Polynomials
by Jeong Gon Lee, Wonjoo Kim and Lee-Chae Jang
Symmetry 2019, 11(5), 709; https://doi.org/10.3390/sym11050709 - 24 May 2019
Viewed by 1710
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)
9 pages, 701 KiB  
Article
The Power Sums Involving Fibonacci Polynomials and Their Applications
by Li Chen and Xiao Wang
Symmetry 2019, 11(5), 635; https://doi.org/10.3390/sym11050635 - 06 May 2019
Cited by 7 | Viewed by 2573
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)
14 pages, 281 KiB  
Article
Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials
by Dae San Kim, Han Young Kim, Dojin Kim and Taekyun Kim
Symmetry 2019, 11(5), 613; https://doi.org/10.3390/sym11050613 - 02 May 2019
Cited by 16 | Viewed by 2117
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)
11 pages, 273 KiB  
Article
Extended Degenerate r-Central Factorial Numbers of the Second Kind and Extended Degenerate r-Central Bell Polynomials
by Dae San Kim, Dmitry V. Dolgy, Taekyun Kim and Dojin Kim
Symmetry 2019, 11(4), 595; https://doi.org/10.3390/sym11040595 - 24 Apr 2019
Cited by 6 | Viewed by 2194
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)
11 pages, 267 KiB  
Article
The Extended Minimax Disparity RIM Quantifier Problem
by Dug Hun Hong
Symmetry 2019, 11(4), 481; https://doi.org/10.3390/sym11040481 - 03 Apr 2019
Viewed by 1355
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)
11 pages, 283 KiB  
Article
The Solution Equivalence to General Models for the RIM Quantifier Problem
by Dug Hun Hong
Symmetry 2019, 11(4), 455; https://doi.org/10.3390/sym11040455 - 01 Apr 2019
Viewed by 1632
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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1 pages, 149 KiB  
Correction
Correction: Kim, T.; Khan, W.A.; Sharma, S.K.; Ghayasuddin, M. A Note on Parametric Kinds of the Degenerate Poly-Bernoulli and Poly-Genocchi Polynomials. Symmetry 2020, 12(4), 614
by Taekyun Kim, Waseem A. Khan, Sunil Kumar Sharma and Mohd Ghayasuddin
Symmetry 2020, 12(6), 871; https://doi.org/10.3390/sym12060871 - 26 May 2020
Viewed by 1257
Abstract
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ) [...] Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
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