Mathematics

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13 pages, 236 KiB

Open AccessArticle

A Parametric Kind of the Degenerate Fubini Numbers and Polynomials

by Sunil Kumar Sharma, Waseem A. Khan and Cheon Seoung Ryoo

Mathematics 2020, 8(3), 405; https://doi.org/10.3390/math8030405 - 12 Mar 2020

Cited by 15 | Viewed by 2378

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)

12 pages, 278 KiB

Open AccessArticle

On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals

by Ahmed Bakhet and Fuli He

Mathematics 2020, 8(2), 232; https://doi.org/10.3390/math8020232 - 10 Feb 2020

Cited by 13 | Viewed by 2544

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)

13 pages, 282 KiB

Open AccessArticle

Fractional Supersymmetric Hermite Polynomials

by Fethi Bouzeffour and Wissem Jedidi

Mathematics 2020, 8(2), 193; https://doi.org/10.3390/math8020193 - 5 Feb 2020

Cited by 4 | Viewed by 2395

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)

8 pages, 1142 KiB

Open AccessCommunication

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

by Pavel Praks and Dejan Brkić

Mathematics 2020, 8(1), 26; https://doi.org/10.3390/math8010026 - 20 Dec 2019

Cited by 4 | Viewed by 2735

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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12 pages, 664 KiB

Open AccessFeature PaperArticle

Iterating the Sum of Möbius Divisor Function and Euler Totient Function

by Daeyeoul Kim, Umit Sarp and Sebahattin Ikikardes

Mathematics 2019, 7(11), 1083; https://doi.org/10.3390/math7111083 - 9 Nov 2019

Cited by 1 | Viewed by 3790

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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11 pages, 1181 KiB

Open AccessArticle

Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations

by Kyung-Won Hwang, Cheon Seoung Ryoo and Nam Soon Jung

Mathematics 2019, 7(8), 736; https://doi.org/10.3390/math7080736 - 12 Aug 2019

Cited by 5 | Viewed by 3328

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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15 pages, 259 KiB

Open AccessArticle

Truncated Fubini Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2019, 7(5), 431; https://doi.org/10.3390/math7050431 - 15 May 2019

Cited by 12 | Viewed by 2900

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)

12 pages, 314 KiB

Open AccessFeature PaperArticle

On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights

by Chunji Li and Cheon Seoung Ryoo

Mathematics 2019, 7(2), 212; https://doi.org/10.3390/math7020212 - 25 Feb 2019

Viewed by 2511

Abstract

Let

1 < a < b < c < d

and

{\hat{α}}_{[5]} : = {(1, \sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{d})}^{\land}

be a weighted sequence that is recursively generated by five weights [...] Read more.

Let

1 < a < b < c < d

and

{\hat{α}}_{[5]} : = {(1, \sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{d})}^{\land}

be a weighted sequence that is recursively generated by five weights

1, \sqrt{a}, \sqrt{b},

\sqrt{c}, \sqrt{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) : \sqrt{x}, {\hat{α}}_{[5]}

and

α (y, x) : \sqrt{y}, \sqrt{x}, {\hat{α}}_{[5]} .

Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications)

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17 pages, 311 KiB

Open AccessArticle

Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

by Huxiao Luo, Shengjun Li and Chunji Li

Mathematics 2019, 7(2), 151; https://doi.org/10.3390/math7020151 - 5 Feb 2019

Cited by 2 | Viewed by 2385

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)

10 pages, 251 KiB

Open AccessArticle

Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

by Taekyun Kim and Dae San Kim

Mathematics 2019, 7(1), 47; https://doi.org/10.3390/math7010047 - 4 Jan 2019

Cited by 6 | Viewed by 2591

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)

11 pages, 340 KiB

Open AccessArticle

Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros

by Cheon Seoung Ryoo

Mathematics 2019, 7(1), 23; https://doi.org/10.3390/math7010023 - 26 Dec 2018

Cited by 8 | Viewed by 3401

Abstract

In this paper, we study differential equations arising from the generating functions of Hermit Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet polynomials. Finally, [...] Read more.

In this paper, we study differential equations arising from the generating functions of Hermit Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet polynomials. Finally, use the computer to view the location of the zeros of Hermite Kamp

\overset{´}{e}

de F

\overset{´}{e}

riet polynomials. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications)

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