Special Issue "Current Trends in Symmetric Polynomials with their Applications"

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

Deadline for manuscript submissions: closed (28 February 2019).

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, S. Korea
Tel. +82-(0)2-940-8368
Interests: integral equations
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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

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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 (23 papers)

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Research

Open AccessArticle
Fluctuation Theorem of Information Exchange between Subsystems that Co-Evolve in Time
Symmetry 2019, 11(3), 433; https://doi.org/10.3390/sym11030433 - 22 Mar 2019
Cited by 1
Abstract
Sagawa and Ueda established a fluctuation theorem of information exchange by revealing the role of correlations in stochastic thermodynamics and unified the non-equilibrium thermodynamics of measurement and feedback control. They considered a process where a non-equilibrium system exchanges information with other degrees of [...] Read more.
Sagawa and Ueda established a fluctuation theorem of information exchange by revealing the role of correlations in stochastic thermodynamics and unified the non-equilibrium thermodynamics of measurement and feedback control. They considered a process where a non-equilibrium system exchanges information with other degrees of freedom such as an observer or a feedback controller. They proved the fluctuation theorem of information exchange under the assumption that the state of the other degrees of freedom that exchange information with the system does not change over time while the states of the system evolve in time. Here we relax this constraint and prove that the same form of the fluctuation theorem holds even if both subsystems co-evolve during information exchange processes. This result may extend the applicability of the fluctuation theorem of information exchange to a broader class of non-equilibrium processes, such as a dynamic coupling in biological systems, where subsystems that exchange information interact with each other. Full article
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Open AccessArticle
A Note on the Sequence Related to Catalan Numbers
Symmetry 2019, 11(3), 371; https://doi.org/10.3390/sym11030371 - 13 Mar 2019
Abstract
The main purpose of this paper is to find explicit expressions for two sequences and to solve two related conjectures arising from the recent study of sums of finite products of Catalan numbers by Zhang and Chen. Full article
Open AccessArticle
Bernoulli Polynomials and Their Some New Congruence Properties
Symmetry 2019, 11(3), 365; https://doi.org/10.3390/sym11030365 - 11 Mar 2019
Cited by 1
Abstract
The aim of this article is to use the fundamental modus and the properties of the Euler polynomials and Bernoulli polynomials to prove some new congruences related to Bernoulli polynomials. One of them is that for any integer h or any non-negative integer [...] Read more.
The aim of this article is to use the fundamental modus and the properties of the Euler polynomials and Bernoulli polynomials to prove some new congruences related to Bernoulli polynomials. One of them is that for any integer h or any non-negative integer n, we obtain the congruence B 2 n + 1 ( 2 h ) 0 mod ( 2 n + 1 ) , where B n ( x ) are Bernoulli polynomials. Full article
Open AccessArticle
Connection Problem for Sums of Finite Products of Legendre and Laguerre Polynomials
Symmetry 2019, 11(3), 317; https://doi.org/10.3390/sym11030317 - 02 Mar 2019
Cited by 5
Abstract
The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi [...] Read more.
The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 . Full article
Open AccessArticle
On Central Complete and Incomplete Bell Polynomials I
Symmetry 2019, 11(2), 288; https://doi.org/10.3390/sym11020288 - 22 Feb 2019
Cited by 5
Abstract
In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as ’central’ analogues for complete and incomplete Bell polynomials. Further, some properties [...] Read more.
In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as ’central’ analogues for complete and incomplete Bell polynomials. Further, some properties and identities for these polynomials are investigated. In particular, we provide explicit formulas for the central complete and incomplete Bell polynomials related to central factorial numbers of the second kind. Full article
Open AccessArticle
New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers
Symmetry 2019, 11(2), 264; https://doi.org/10.3390/sym11020264 - 20 Feb 2019
Cited by 1
Abstract
In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials [...] Read more.
In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found. Full article
Open AccessArticle
A Modified PML Acoustic Wave Equation
Symmetry 2019, 11(2), 177; https://doi.org/10.3390/sym11020177 - 02 Feb 2019
Abstract
In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable [...] Read more.
In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers. Full article
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Open AccessArticle
On the Catalan Numbers and Some of Their Identities
Symmetry 2019, 11(1), 62; https://doi.org/10.3390/sym11010062 - 08 Jan 2019
Cited by 1
Abstract
The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the [...] Read more.
The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers. Full article
Open AccessArticle
Representation by Chebyshev Polynomials for Sums of Finite Products of Chebyshev Polynomials
Symmetry 2018, 10(12), 742; https://doi.org/10.3390/sym10120742 - 11 Dec 2018
Cited by 3
Abstract
In this paper, we consider sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the previously-studied ones. We represent each of them as linear combinations of Chebyshev polynomials of all kinds whose coefficients involve [...] Read more.
In this paper, we consider sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the previously-studied ones. We represent each of them as linear combinations of Chebyshev polynomials of all kinds whose coefficients involve some terminating hypergeometric functions 2 F 1 . The results may be viewed as a generalization of the linearization problem, which is concerned with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. These representations are obtained by explicit computations. Full article
Open AccessArticle
Symmetry Identities of Changhee Polynomials of Type Two
Symmetry 2018, 10(12), 740; https://doi.org/10.3390/sym10120740 - 11 Dec 2018
Cited by 1
Abstract
In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry [...] Read more.
In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry for the fermionic p-adic integral on Z p . Full article
Open AccessArticle
Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ
Symmetry 2018, 10(12), 675; https://doi.org/10.3390/sym10120675 - 29 Nov 2018
Abstract
We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, [...] Read more.
We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities. Full article
Open AccessArticle
A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications
Symmetry 2018, 10(11), 652; https://doi.org/10.3390/sym10110652 - 19 Nov 2018
Abstract
The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials [...] Read more.
The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations. Full article
Open AccessFeature PaperArticle
Symmetric Properties of Carlitz’s Type q-Changhee Polynomials
Symmetry 2018, 10(11), 634; https://doi.org/10.3390/sym10110634 - 13 Nov 2018
Abstract
Changhee polynomials were introduced by Kim, and the generalizations of these polynomials have been characterized. In our paper, we investigate various interesting symmetric identities for Carlitz’s type q-Changhee polynomials under the symmetry group of order n arising from the fermionic p-adic [...] Read more.
Changhee polynomials were introduced by Kim, and the generalizations of these polynomials have been characterized. In our paper, we investigate various interesting symmetric identities for Carlitz’s type q-Changhee polynomials under the symmetry group of order n arising from the fermionic p-adic q-integral on Z p . Full article
Open AccessArticle
On Classical Gauss Sums and Some of Their Properties
Symmetry 2018, 10(11), 625; https://doi.org/10.3390/sym10110625 - 11 Nov 2018
Cited by 2
Abstract
The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it. Full article
Open AccessArticle
Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds
Symmetry 2018, 10(11), 617; https://doi.org/10.3390/sym10110617 - 09 Nov 2018
Cited by 3
Abstract
This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations [...] Read more.
This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 . Full article
Open AccessArticle
Fibonacci and Lucas Numbers of the Form 2a + 3b + 5c + 7d
Symmetry 2018, 10(10), 509; https://doi.org/10.3390/sym10100509 - 16 Oct 2018
Cited by 1
Abstract
In this paper, we find all Fibonacci and Lucas numbers written in the form 2 a + 3 b + 5 c + 7 d , in non-negative integers a , b , c , d , with 0 max { a [...] Read more.
In this paper, we find all Fibonacci and Lucas numbers written in the form 2 a + 3 b + 5 c + 7 d , in non-negative integers a , b , c , d , with 0 max { a , b , c } d . Full article
Open AccessArticle
A Note on Modified Degenerate Gamma and Laplace Transformation
Symmetry 2018, 10(10), 471; https://doi.org/10.3390/sym10100471 - 10 Oct 2018
Cited by 1
Abstract
Kim-Kim studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. In this paper, we define modified degenerate gamma and modified degenerate Laplace transformation and investigate some properties and formulas related to them. Full article
Open AccessArticle
On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials
Symmetry 2018, 10(10), 451; https://doi.org/10.3390/sym10100451 - 01 Oct 2018
Cited by 2
Abstract
The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties [...] Read more.
The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found. Full article
Open AccessArticle
A New Sequence and Its Some Congruence Properties
Symmetry 2018, 10(9), 359; https://doi.org/10.3390/sym10090359 - 24 Aug 2018
Cited by 2
Abstract
The aim of this paper is to study the congruence properties of a new sequence, which is closely related to Fubini polynomials and Euler numbers, using the elementary method and the properties of the second kind Stirling numbers. As results, we obtain some [...] Read more.
The aim of this paper is to study the congruence properties of a new sequence, which is closely related to Fubini polynomials and Euler numbers, using the elementary method and the properties of the second kind Stirling numbers. As results, we obtain some interesting congruences for it. This solves a problem proposed in a published paper. Full article
Open AccessArticle
On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials
Symmetry 2018, 10(8), 311; https://doi.org/10.3390/sym10080311 - 01 Aug 2018
Cited by 3
Abstract
We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the [...] Read more.
We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers. Full article
Open AccessArticle
Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers
Symmetry 2018, 10(8), 303; https://doi.org/10.3390/sym10080303 - 01 Aug 2018
Cited by 9
Abstract
The aim of this paper is to use elementary methods and the recursive properties of a special sequence to study the computational problem of one kind symmetric sums involving Fubini polynomials and Euler numbers, and give an interesting computational formula for it. At [...] Read more.
The aim of this paper is to use elementary methods and the recursive properties of a special sequence to study the computational problem of one kind symmetric sums involving Fubini polynomials and Euler numbers, and give an interesting computational formula for it. At the same time, we also give a recursive calculation method for the general case. Full article
Open AccessArticle
Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials
Symmetry 2018, 10(7), 258; https://doi.org/10.3390/sym10070258 - 03 Jul 2018
Cited by 10
Abstract
Here, we consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then, we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials, which involve the hypergeometric function 3 [...] Read more.
Here, we consider the sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then, we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials, which involve the hypergeometric function 3F2. Full article
Open AccessArticle
Symmetric Identities for Fubini Polynomials
Symmetry 2018, 10(6), 219; https://doi.org/10.3390/sym10060219 - 14 Jun 2018
Cited by 9
Abstract
We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, representing generating functions of three w-torsion [...] Read more.
We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials. Full article
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