Special Issue "Polynomials: Theory and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 13587

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Special Issue Editor

Prof. Dr. Cheon-Seoung Ryoo
E-Mail Website
Guest Editor
Department of Mathematics, Hannam University, Daejeon 34430, Korea
Interests: numerical analysis; differential and difference equations; scientific computing; dynamical systems; quantum calculus; special functions
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Special Issue Information

Dear Colleagues,

The importance of polynomials in the interdisciplinary field of mathematics, engineering, and science is well known. Over the past several decades, research on polynomials has been conducted extensively in many disciplines.

This Special Issue welcomes all research papers related to polynomials in mathematics, science, and industry.

Potential topics include but are not limited to the following:

  • The modern umbral calculus (binomial, Appell, and Sheffer polynomial sequences)
  • Orthogonal polynomials, matrix orthogonal polynomials, multiple orthogonal polynomials
  • Matrix and determinant approach to special polynomial sequences
  • Applications of special polynomial sequences
  • Number theory and special functions
  • Asymptotic methods in orthogonal polynomials
  • Fractional calculus and special functions
  • Symbolic computations and special functions
  • Applications of special functions to statistics, physical sciences, and engineering

Prof. Dr. Cheon-Seoung Ryoo
Guest Editor

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Keywords

  • umbral calculus
  • orthogonal polynomials, matrix orthogonal polynomials
  • special polynomial sequences
  • applications of special polynomial sequences
  • number theory and special functions
  • fractional calculus and special functions
  • symbolic computations and special functions
  • applications of special functions to statistics, physical sciences, and engineering

Published Papers (11 papers)

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Research

Article
A Parametric Kind of the Degenerate Fubini Numbers and Polynomials
Mathematics 2020, 8(3), 405; https://doi.org/10.3390/math8030405 - 12 Mar 2020
Cited by 10 | Viewed by 979
Abstract
In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of [...] Read more.
In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling numbers. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Article
On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals
Mathematics 2020, 8(2), 232; https://doi.org/10.3390/math8020232 - 10 Feb 2020
Cited by 11 | Viewed by 1036
Abstract
In this paper, we first introduce the 2-variables Konhauser matrix polynomials; then, we investigate some properties of these matrix polynomials such as generating matrix relations, integral representations, and finite sum formulae. Finally, we obtain the fractional integrals of the 2-variables Konhauser matrix polynomials. [...] Read more.
In this paper, we first introduce the 2-variables Konhauser matrix polynomials; then, we investigate some properties of these matrix polynomials such as generating matrix relations, integral representations, and finite sum formulae. Finally, we obtain the fractional integrals of the 2-variables Konhauser matrix polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Article
Fractional Supersymmetric Hermite Polynomials
Mathematics 2020, 8(2), 193; https://doi.org/10.3390/math8020193 - 05 Feb 2020
Cited by 1 | Viewed by 927
Abstract
We provide a realization of fractional supersymmetry quantum mechanics of order r, where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be [...] Read more.
We provide a realization of fractional supersymmetry quantum mechanics of order r, where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be expressed in terms of the associated Laguerre orthogonal polynomials and have shown that their zeros are the eigenvalues of the Hermitian supercharge. We call them the supersymmetric generalized Hermite polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Communication
Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation
Mathematics 2020, 8(1), 26; https://doi.org/10.3390/math8010026 - 20 Dec 2019
Cited by 4 | Viewed by 1123
Abstract
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use [...] Read more.
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
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Article
Iterating the Sum of Möbius Divisor Function and Euler Totient Function
Mathematics 2019, 7(11), 1083; https://doi.org/10.3390/math7111083 - 09 Nov 2019
Cited by 1 | Viewed by 1288
Abstract
In this paper, according to some numerical computational evidence, we investigate and prove certain identities and properties on the absolute Möbius divisor functions and Euler totient function when they are iterated. Subsequently, the relationship between the absolute Möbius divisor function with Fermat primes [...] Read more.
In this paper, according to some numerical computational evidence, we investigate and prove certain identities and properties on the absolute Möbius divisor functions and Euler totient function when they are iterated. Subsequently, the relationship between the absolute Möbius divisor function with Fermat primes has been researched and some results have been obtained. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
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Article
Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations
Mathematics 2019, 7(8), 736; https://doi.org/10.3390/math7080736 - 12 Aug 2019
Cited by 3 | Viewed by 1174
Abstract
In this paper, we study differential equations arising from the generating function of the ( r , β ) -Bell polynomials. We give explicit identities for the ( r , β ) -Bell polynomials. Finally, we find the zeros of the [...] Read more.
In this paper, we study differential equations arising from the generating function of the ( r , β ) -Bell polynomials. We give explicit identities for the ( r , β ) -Bell polynomials. Finally, we find the zeros of the ( r , β ) -Bell equations with numerical experiments. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
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Article
Truncated Fubini Polynomials
Mathematics 2019, 7(5), 431; https://doi.org/10.3390/math7050431 - 15 May 2019
Cited by 7 | Viewed by 1222
Abstract
In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling [...] Read more.
In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Article
On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights
Mathematics 2019, 7(2), 212; https://doi.org/10.3390/math7020212 - 25 Feb 2019
Cited by 1 | Viewed by 1062
Abstract
Let 1 < a < b < c < d and α ^ 5 : = 1 , a , b , c , d be a weighted sequence that is recursively generated by five weights [...] Read more.
Let 1 < a < b < c < d and α ^ 5 : = 1 , a , b , c , d be a weighted sequence that is recursively generated by five weights 1 , a , b , c , d . In this paper, we give sufficient conditions for the positive quadratic hyponormalities of W α x and W α y , x , with α x : x , α ^ 5 and α y , x : y , x , α ^ 5 . Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
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Article
Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity
Mathematics 2019, 7(2), 151; https://doi.org/10.3390/math7020151 - 05 Feb 2019
Cited by 1 | Viewed by 1155
Abstract
In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. [...] Read more.
In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016). Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Article
Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials
Mathematics 2019, 7(1), 47; https://doi.org/10.3390/math7010047 - 04 Jan 2019
Cited by 4 | Viewed by 1187
Abstract
In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate [...] Read more.
In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
Article
Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros
Mathematics 2019, 7(1), 23; https://doi.org/10.3390/math7010023 - 26 Dec 2018
Cited by 5 | Viewed by 1374
Abstract
In this paper, we study differential equations arising from the generating functions of Hermit Kamp e ´ de F e ´ riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp e ´ de F e ´ riet polynomials. Finally, [...] Read more.
In this paper, we study differential equations arising from the generating functions of Hermit Kamp e ´ de F e ´ riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp e ´ de F e ´ riet polynomials. Finally, use the computer to view the location of the zeros of Hermite Kamp e ´ de F e ´ riet polynomials. Full article
(This article belongs to the Special Issue Polynomials: Theory and Applications)
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