Research

18 pages, 1242 KiB

Open AccessArticle

Novel Properties of q-Sine-Based and q-Cosine-Based q-Fubini Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi, Cheon Seoung Ryoo and Ugur Duran

Symmetry 2023, 15(2), 356; https://doi.org/10.3390/sym15020356 - 28 Jan 2023

Cited by 5 | Viewed by 788

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling [...] Read more.

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling numbers of the second kind are derived. Moreover, some approximate zeros of the q-sinebased and q-cosine-Based q-Fubini polynomials in a complex plane are examined, and lastly, these zeros are shown using figures. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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10 pages, 291 KiB

Open AccessArticle

New Results on Integral Operator for a Subclass of Analytic Functions Using Differential Subordinations and Superordinations

by Fatima Obaid Salman and Waggas Galib Atshan

Symmetry 2023, 15(2), 295; https://doi.org/10.3390/sym15020295 - 20 Jan 2023

Cited by 1 | Viewed by 845

Abstract

In this paper, we discuss and introduce a new study using an integral operator

w_{k, μ}^{m}

in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In [...] Read more.

In this paper, we discuss and introduce a new study using an integral operator

w_{k, μ}^{m}

in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory’s features and outcomes are symmetric to those derived using the differential subordination theory. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

21 pages, 1718 KiB

Open AccessArticle

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

by Waleed Mohamed Abd-Elhameed, Seraj Omar Alkhamisi, Amr Kamel Amin and Youssri Hassan Youssri

Symmetry 2023, 15(1), 138; https://doi.org/10.3390/sym15010138 - 03 Jan 2023

Cited by 3 | Viewed by 861

Abstract

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev [...] Read more.

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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22 pages, 390 KiB

Open AccessFeature PaperArticle

Symmetries of Non-Linear ODEs: Lambda Extensions of the Ising Correlations

by Salah Boukraa and Jean-Marie Maillard

Symmetry 2022, 14(12), 2622; https://doi.org/10.3390/sym14122622 - 11 Dec 2022

Viewed by 748

Abstract

This paper provides several illustrations of the numerous remarkable properties of the lambda extensions of the two-point correlation functions of the Ising model, shedding some light on the non-linear ODEs of the Painlevé type they satisfy. We first show that this concept also [...] Read more.

This paper provides several illustrations of the numerous remarkable properties of the lambda extensions of the two-point correlation functions of the Ising model, shedding some light on the non-linear ODEs of the Painlevé type they satisfy. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples, namely

C (0, 5)

and

C (2, 5)

at

ν = - k

. We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda extensions: for an infinite set of (algebraic) values of

λ

these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind

K

and

E

. For generic values of

λ

these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of

λ

these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlevé VI remains an extremely difficult challenge. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

20 pages, 506 KiB

Open AccessArticle

Qualitative Analysis in a Beddington–DeAngelis Type Predator–Prey Model with Two Time Delays

by Miao Peng, Rui Lin, Yue Chen, Zhengdi Zhang and Mostafa M. A. Khater

Symmetry 2022, 14(12), 2535; https://doi.org/10.3390/sym14122535 - 30 Nov 2022

Cited by 2 | Viewed by 1300

Abstract

In this paper, we investigate a delayed predator–prey model with a prey refuge where the predator population eats the prey according to the Beddington–DeAngelis type functional response. Firstly, we consider the existence of equilibrium points. By analyzing the corresponding characteristic equations, the local [...] Read more.

In this paper, we investigate a delayed predator–prey model with a prey refuge where the predator population eats the prey according to the Beddington–DeAngelis type functional response. Firstly, we consider the existence of equilibrium points. By analyzing the corresponding characteristic equations, the local stability of the trivial equilibrium, the predator–extinction balance, and the coexistence equilibrium of the system are discussed, and the existence of Hopf bifurcations concerning both delays at the coexistence equilibrium are established. Then, in accordance with the standard form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and stability of bifurcating period solutions are derived. Finally, representative numerical simulations are performed to validate the theoretical analysis. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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11 pages, 951 KiB

Open AccessArticle

A Computational Scheme for the Numerical Results of Time-Fractional Degasperis–Procesi and Camassa–Holm Models

by Muhammad Nadeem, Hossein Jafari, Ali Akgül and Manuel De la Sen

Symmetry 2022, 14(12), 2532; https://doi.org/10.3390/sym14122532 - 30 Nov 2022

Cited by 2 | Viewed by 944

Abstract

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (

L

T) and homotopy perturbation method (HPM) to formulate the [...] Read more.

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (

L

T) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (

L

HPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed scheme. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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0 pages, 322 KiB

Open AccessArticle

On Some Asymptotic Expansions for the Gamma Function

by Mansour Mahmoud and Hanan Almuashi

Symmetry 2022, 14(11), 2459; https://doi.org/10.3390/sym14112459 - 20 Nov 2022

Cited by 3 | Viewed by 1442

Abstract

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the [...] Read more.

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function

Γ (1 + w) \sim \sqrt{2 π w} {(w / e)}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2} exp (\sum_{r = 1}^{\infty}, \frac{μ_{r}}{w^{r}}), w \to \infty

, with the recurrence relation of coefficients

μ_{r}

. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of

Γ (w)

better than some of its recent ones. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

25 pages, 384 KiB

Open AccessArticle

Some Formulas and Recurrences of Certain Orthogonal Polynomials Generalizing Chebyshev Polynomials of the Third-Kind

by Waleed Mohamed Abd-Elhameed and Mohamed Salem Al-Harbi

Symmetry 2022, 14(11), 2309; https://doi.org/10.3390/sym14112309 - 03 Nov 2022

Cited by 2 | Viewed by 1175

Abstract

This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal polynomials called Chebyshev polynomials of the third-kind. Some new formulas are developed for these polynomials. We will show that some of the previous results in the literature can [...] Read more.

This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal polynomials called Chebyshev polynomials of the third-kind. Some new formulas are developed for these polynomials. We will show that some of the previous results in the literature can be considered special ones of our derived formulas. The derivatives of the moments of these polynomials are derived. Hence, two important formulas that explicitly give the derivatives and the moments of these polynomials in terms of their original ones can be deduced as special cases. Some new expressions for the derivatives of different symmetric and non-symmetric polynomials are expressed as combinations of the generalized third-kind Chebyshev polynomials. Some new linearization formulas are also given using different approaches. Some of the appearing coefficients in derivatives and linearization formulas are given in terms of different hypergeometric functions. Furthermore, in several cases, the existing hypergeometric functions can be summed using some standard formulas in the literature or through the employment of suitable symbolic algebra, in particular, Zeilberger’s algorithm. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

14 pages, 6936 KiB

Open AccessArticle

Mapping Properties of Associate Laguerre Polynomials in Leminiscate, Exponential and Nephroid Domain

by Saiful R. Mondal

Symmetry 2022, 14(11), 2303; https://doi.org/10.3390/sym14112303 - 03 Nov 2022

Cited by 1 | Viewed by 1487

Abstract

The function

P_{L} (z) = \sqrt{1 + z}

maps the unit disc

D = {z \in C : | z | < 1}

to a leminscate which is symmetric about the x-axis. The conditions on the parameters

α

[...] Read more.

The function

P_{L} (z) = \sqrt{1 + z}

maps the unit disc

D = {z \in C : | z | < 1}

to a leminscate which is symmetric about the x-axis. The conditions on the parameters

α

and n, for which the associated Laguerre polynomial (ALP)

L_{n}^{α}

maps unit disc into the leminscate domain, are deduced in this article. We also establish the condition under which a function involving

L_{n}^{α}

maps

D

to a domain subordinated by

ϕ_{N_{e}} (z) = 1 - z + z^{3} / 3

,

ϕ_{e} (z) = e^{z}

, and

ϕ_{A} (z) = 1 + A z

,

A \in (0, 1]

. We provide several graphical presentations for a clear view of some of the obtained results. The possibilities for the improvements of the results are also highlighted. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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25 pages, 372 KiB

Open AccessArticle

Novel Identities of Bernoulli Polynomials Involving Closed Forms for Some Definite Integrals

by Waleed Mohamed Abd-Elhameed and Amr Kamel Amin

Symmetry 2022, 14(11), 2284; https://doi.org/10.3390/sym14112284 - 31 Oct 2022

Cited by 3 | Viewed by 1106

Abstract

This paper presents new results of Bernoulli polynomials. New derivative expressions of some celebrated orthogonal polynomials and other polynomials are given in terms of Bernoulli polynomials. Hence, some new connection formulas between these polynomials and Bernoulli polynomials are also deduced. The linking coefficients [...] Read more.

This paper presents new results of Bernoulli polynomials. New derivative expressions of some celebrated orthogonal polynomials and other polynomials are given in terms of Bernoulli polynomials. Hence, some new connection formulas between these polynomials and Bernoulli polynomials are also deduced. The linking coefficients involve hypergeometric functions of different arguments that can be summed in some cases. Formulas that express some celebrated numbers in terms of Bernoulli numbers are displayed. Based on the new connection formulas between different polynomials and Bernoulli polynomials, along with some well-known integrals involving these polynomials, new closed forms for some definite integrals are given. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

16 pages, 973 KiB

Open AccessArticle

Sandwich Theorems for a New Class of Complete Homogeneous Symmetric Functions by Using Cyclic Operator

by Intissar Abdulhur Kadum, Waggas Galib Atshan and Areej Tawfeeq Hameed

Symmetry 2022, 14(10), 2223; https://doi.org/10.3390/sym14102223 - 21 Oct 2022

Cited by 2 | Viewed by 1296

Abstract

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric [...] Read more.

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

14 pages, 777 KiB

Open AccessArticle

Perturbed Newton Methods for Solving Nonlinear Equations with Applications

by Ioannis K. Argyros, Samundra Regmi, Stepan Shakhno and Halyna Yarmola

Symmetry 2022, 14(10), 2206; https://doi.org/10.3390/sym14102206 - 20 Oct 2022

Cited by 4 | Viewed by 1137

Abstract

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed [...] Read more.

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results involving the derivative only appearing on these methods. Hence, we have expanded the usage of these methods. In the case of the Jarratt-like family of methods, our results also hold for Banach space-valued equations. Moreover, we compare the convergence ball and the dynamical features both theoretically and in numerical experiments. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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13 pages, 293 KiB

Open AccessArticle

On the Solutions of Quaternion Difference Equations in Terms of Generalized Fibonacci-Type Numbers

by Kübra Gül

Symmetry 2022, 14(10), 2190; https://doi.org/10.3390/sym14102190 - 18 Oct 2022

Viewed by 1275

Abstract

The aim of this paper is to investigate the solution of the following difference equation

z_{n + 1} = {(p_{n})}^{- 1}, n \in N_{0}, N_{0} = N \cup \{0\}

where [...] Read more.

The aim of this paper is to investigate the solution of the following difference equation

z_{n + 1} = {(p_{n})}^{- 1}, n \in N_{0}, N_{0} = N \cup \{0\}

where

p_{n} = a + b z_{n} + c z_{n - 1} z_{n}

with the parameters a, b, c and the initial values

z_{- 1}, z_{0}

are nonzero quaternions such that their solutions are associated with generalized Fibonacci-type numbers. Furthermore, we deal with the solutions to the following symmetric system of difference equations given by

z_{n + 1} = {(q_{n})}^{- 1}, w_{n + 1} = {(r_{n})}^{- 1}, n \in N_{0}

where

q_{n} =

a + b w_{n} + c z_{n - 1} w_{n}

and

r_{n} = a + b z_{n} + c w_{n - 1} z_{n}

. We provide the solution to the third-order quaternion linear difference equation in terms of the zeros of the characteristic polynomial associated with the linear difference equation. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

14 pages, 282 KiB

Open AccessArticle

A Note on Degenerate Catalan-Daehee Numbers and Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi and Ugur Duran

Symmetry 2022, 14(10), 2169; https://doi.org/10.3390/sym14102169 - 16 Oct 2022

Cited by 3 | Viewed by 972

Abstract

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

λ

-Daehee numbers, Stirling numbers [...] Read more.

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

λ

-Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan–Daehee numbers and Stirling numbers of the second kind. In addition, we prove an implicit summation formula and a symmetric identity, and we derive an explicit expression for the degenerate Catalan–Daehee polynomials including the Stirling numbers of the first kind and Bernoulli polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

16 pages, 334 KiB

Open AccessArticle

Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

by Alina Alb Lupaş and Georgia Irina Oros

Symmetry 2022, 14(10), 2097; https://doi.org/10.3390/sym14102097 - 08 Oct 2022

Cited by 1 | Viewed by 987

Abstract

Studies regarding the two dual notions are conducted in this paper using Riemann–Liouville fractional integral of q-hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and fuzzy best subordinants are given in the theorems investigating fuzzy differential subordinations [...] Read more.

Studies regarding the two dual notions are conducted in this paper using Riemann–Liouville fractional integral of q-hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and fuzzy best subordinants are given in the theorems investigating fuzzy differential subordinations and superordinations, respectively. Moreover, corollaries are stated by considering particular functions with known geometric properties as fuzzy best dominant and fuzzy best subordinant in the proved results. The study is completed by stating fuzzy differential sandwich theorems followed by related corollaries combining the results previously established concerning fuzzy differential subordinations and superordinations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

29 pages, 590 KiB

Open AccessArticle

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

by Djihane Benmorsli, Zine El Abiddine Fellah, Djema Belgroune, Nicholas O. Ongwen, Erick Ogam, Claude Depollier and Mohamed Fellah

Symmetry 2022, 14(10), 1971; https://doi.org/10.3390/sym14101971 - 21 Sep 2022

Cited by 2 | Viewed by 1266

Abstract

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian [...] Read more.

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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13 pages, 273 KiB

Open AccessArticle

Fourier Series Expansion and Integral Representation of Apostol-Type Frobenius–Euler Polynomials of Complex Parameters and Order α

by Cristina Corcino, Roberto Corcino and Jeremar Casquejo

Symmetry 2022, 14(9), 1860; https://doi.org/10.3390/sym14091860 - 06 Sep 2022

Viewed by 1203

Abstract

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order

α

are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type [...] Read more.

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order

α

are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials resulting in their expansions and integral representations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

9 pages, 260 KiB

Open AccessArticle

Subordination Results on the q-Analogue of the Sălăgean Differential Operator

by Alina Alb Lupaş

Symmetry 2022, 14(8), 1744; https://doi.org/10.3390/sym14081744 - 22 Aug 2022

Cited by 9 | Viewed by 1116

Abstract

Aspects related to applications in the geometric function theory of q-calculus are presented in this paper. The study concerns the investigation of certain q-analogue differential operators in order to obtain their geometrical properties, which could be further developed in studies. Several [...] Read more.

Aspects related to applications in the geometric function theory of q-calculus are presented in this paper. The study concerns the investigation of certain q-analogue differential operators in order to obtain their geometrical properties, which could be further developed in studies. Several interesting properties of the q-analogue of the Sălăgean differential operator are obtained here by using the differential subordination method. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

18 pages, 1286 KiB

Open AccessArticle

Explicit Properties of q-Cosine and q-Sine Array-Type Polynomials Containing Symmetric Structures

by Maryam Salem Alatawi, Waseem Ahmad Khan and Cheon Seoung Ryoo

Symmetry 2022, 14(8), 1675; https://doi.org/10.3390/sym14081675 - 12 Aug 2022

Cited by 2 | Viewed by 1002

Abstract

The main aim of this study is to define parametric kinds of

λ

-Array-type polynomials by using q-trigonometric polynomials and to investigate some of their analytical properties and applications. For this purpose, many formulas and relations for these polynomials, including some implicit [...] Read more.

The main aim of this study is to define parametric kinds of

λ

-Array-type polynomials by using q-trigonometric polynomials and to investigate some of their analytical properties and applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules, and relations with the earlier polynomials by utilizing some series manipulation method are derived. Additionally, as an application, the zero values of q-Array-type polynomials are presented by the tables and multifarious graphical representations for these zero values are drawn. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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12 pages, 670 KiB

Open AccessArticle

New Results on Higher-Order Differential Subordination and Superordination for Univalent Analytic Functions Using a New Operator

by Sarab Dakhil Theyab, Waggas Galib Atshan, Alina Alb Lupaș and Habeeb Kareem Abdullah

Symmetry 2022, 14(8), 1576; https://doi.org/10.3390/sym14081576 - 31 Jul 2022

Cited by 6 | Viewed by 1042

Abstract

We present several new results for higher-order (fourth-order) differential subordination and superordination in this paper by using the new operator H_α_,β,_δ,ξ,γ,nf(v) and offer numerous new findings for fourth-order [...] Read more.

We present several new results for higher-order (fourth-order) differential subordination and superordination in this paper by using the new operator H_α_,β,_δ,ξ,γ,nf(v) and offer numerous new findings for fourth-order differential subordination and superordination. The innovative discoveries presented here are connected to those mentioned in previous articles. The differential subordination theory’s characteristics and outcomes are symmetric to the properties gained. Sandwich-type theorems are created by merging differential superordination theory with sandwich-type theorems. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

19 pages, 316 KiB

Open AccessArticle

Extended Ohtsuka–Vălean Sums

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(8), 1551; https://doi.org/10.3390/sym14081551 - 28 Jul 2022

Viewed by 1054

Abstract

The Ohtsuka–Vălean sum is extended to evaluate an extensive number of trigonometric and hyperbolic sums and products. The sums are taken over finite and infinite domains defined in terms of the Hurwitz–Lerch zeta function, which can be simplified to composite functions in special [...] Read more.

The Ohtsuka–Vălean sum is extended to evaluate an extensive number of trigonometric and hyperbolic sums and products. The sums are taken over finite and infinite domains defined in terms of the Hurwitz–Lerch zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products and sums of tangent, cotangent, hyperbolic tangent and hyperbolic cotangent functions, in certain cases raised to a complex number power. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

15 pages, 369 KiB

Open AccessArticle

Analytical Properties of Degenerate Genocchi Polynomials of the Second Kind and Some of Their Applications

by Waseem Ahmad Khan and Maryam Salem Alatawi

Symmetry 2022, 14(8), 1500; https://doi.org/10.3390/sym14081500 - 22 Jul 2022

Cited by 4 | Viewed by 1001

Abstract

The main aim of this study is to define degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and to investigate some of their analytical properties and some applications. For this purpose, many formulas and relations for these polynomials, [...] Read more.

The main aim of this study is to define degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and to investigate some of their analytical properties and some applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules and correlations with the earlier polynomials by utilizing some series manipulation methods, are derived. Additionally, as an application, the zero values of degenerate Genocchi polynomials and numbers of the second kind are presented in tables and multifarious graphical representations for these zero values are shown. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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9 pages, 259 KiB

Open AccessArticle

Closed-Form Derivations of Infinite Sums and Products Involving Trigonometric Functions

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(7), 1418; https://doi.org/10.3390/sym14071418 - 11 Jul 2022

Viewed by 1384

Abstract

We derive a closed-form expression for the infinite sum of the Hurwitz–Lerch zeta function using contour integration. This expression is used to evaluate infinite sum and infinite product formulae involving trigonometric functions expressed in terms of fundamental constants. These types of infinite sums [...] Read more.

We derive a closed-form expression for the infinite sum of the Hurwitz–Lerch zeta function using contour integration. This expression is used to evaluate infinite sum and infinite product formulae involving trigonometric functions expressed in terms of fundamental constants. These types of infinite sums and products have previously been and are currently studied by many mathematicians including Leonhard Euler. The results presented in this paper extend previous work by squaring parameters in the infinite sum of the Hurwitz–Lerch zeta function. This formula allows for new derivations featuring trigonometric functions with angles of powers of 2. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. A table of infinite products is produced highlighting the usefulness of this work and for easy reading by researchers interested in such formulae. Mathematica software was used in assisting with the numerical verification of the results in the tables produced. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

13 pages, 307 KiB

Open AccessArticle

An Asymptotic Expansion for the Generalized Gamma Function

by Mansour Mahmoud, Hanan Almuashi and Hesham Moustafa

Symmetry 2022, 14(7), 1412; https://doi.org/10.3390/sym14071412 - 09 Jul 2022

Cited by 1 | Viewed by 1096

Abstract

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function

Γ_{μ} (v)

and study the completely monotonic (CM) property of a function [...] Read more.

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function

Γ_{μ} (v)

and study the completely monotonic (CM) property of a function involving

Γ_{μ} (v)

and the generalized digamma function

ψ_{μ} (v)

. As a consequence, we establish some bounds for

Γ_{μ} (v)

,

ψ_{μ} (v)

and polygamma functions

ψ_{μ}^{(r)} (v)

,

r \geq 1

. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

33 pages, 470 KiB

Open AccessFeature PaperArticle

Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

by Youssef Abdelaziz, Salah Boukraa, Christoph Koutschan and Jean-Marie Maillard

Symmetry 2022, 14(7), 1297; https://doi.org/10.3390/sym14071297 - 22 Jun 2022

Cited by 2 | Viewed by 1507

Abstract

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables

x

,

y

, and

z

, using creative telescoping, yielding modular forms expressed as pullbacked

_{2} F_{1}

hypergeometric functions, [...] Read more.

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables

x

,

y

, and

z

, using creative telescoping, yielding modular forms expressed as pullbacked

_{2} F_{1}

hypergeometric functions, can be obtained much more efficiently by calculating the

j

-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product

p = x y z

. In other cases where creative telescoping yields pullbacked

_{2} F_{1}

hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked

_{2} F_{1}

hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

16 pages, 374 KiB

Open AccessArticle

Sequence Spaces and Spectrum of q-Difference Operator of Second Order

by Abdullah Alotaibi, Taja Yaying and Syed Abdul Mohiuddine

Symmetry 2022, 14(6), 1155; https://doi.org/10.3390/sym14061155 - 03 Jun 2022

Cited by 6 | Viewed by 1541

Abstract

The sequence spaces

ℓ_{p} (\nabla_{q}^{2})

(0 \leq p < \infty)

and

ℓ_{\infty} (\nabla_{q}^{2})

are introduced by using the q-difference operator

\nabla_{q}^{2}

of the second order. Apart from studying [...] Read more.

The sequence spaces

ℓ_{p} (\nabla_{q}^{2})

(0 \leq p < \infty)

and

ℓ_{\infty} (\nabla_{q}^{2})

are introduced by using the q-difference operator

\nabla_{q}^{2}

of the second order. Apart from studying some basic properties of these spaces, we construct the basis and obtain the

α

-,

β

- and

γ

-duals of these spaces. Besides some matrix classes involving q-difference sequence spaces,

ℓ_{p} (\nabla_{q}^{2})

and

ℓ_{\infty} (\nabla_{q}^{2})

are characterized. The final section is devoted to classifying the spectrum of the q-difference operator

\nabla_{q}^{2}

over the space

ℓ_{1}

of absolutely summable sequences. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

10 pages, 256 KiB

Open AccessArticle

A Note on q-analogue of Degenerate Catalan Numbers Associated with p-adic Integral on

Z_{p}

by Waseem A. Khan

Symmetry 2022, 14(6), 1119; https://doi.org/10.3390/sym14061119 - 29 May 2022

Cited by 12 | Viewed by 1070

Abstract

In this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help of a fermionic p-adic q-integrals on

Z_{p}

and establish some new connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, [...] Read more.

In this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help of a fermionic p-adic q-integrals on

Z_{p}

and establish some new connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, we also find a few new identities and results of this type of polynomials and numbers. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

19 pages, 543 KiB

Open AccessArticle

Impact of Newtonian Heating via Fourier and Fick’s Laws on Thermal Transport of Oldroyd-B Fluid by Using Generalized Mittag-Leffler Kernel

by Chunxia Chen, Aziz Ur Rehman, Muhammad Bilal Riaz, Fahd Jarad and Xiang-E Sun

Symmetry 2022, 14(4), 766; https://doi.org/10.3390/sym14040766 - 07 Apr 2022

Cited by 13 | Viewed by 1461

Abstract

In this manuscript, a new approach to study the fractionalized Oldroyd-B fluid flow based on the fundamental symmetry is described by critically examining the Prabhakar fractional derivative near an infinitely vertical plate, wall slip condition on temperature along with Newtonian heating effects and [...] Read more.

In this manuscript, a new approach to study the fractionalized Oldroyd-B fluid flow based on the fundamental symmetry is described by critically examining the Prabhakar fractional derivative near an infinitely vertical plate, wall slip condition on temperature along with Newtonian heating effects and constant concentration. The phenomenon has been described in forms of partial differential equations along with heat and mass transportation effect taken into account. The Prabhakar fractional operator which was recently introduced is used in this work together with generalized Fick’s and Fourier’s law. The fractional model is transfromed into a non-dimentional form by using some suitable quantities and the symmetry of fluid flow is analyzed. The non-dimensional developed fractional model for momentum, thermal and diffusion equations based on Prabhakar fractional operator has been solved analytically via Laplace transformation method and calculated solutions expressed in terms of Mittag-Leffler special functions. Graphical demonstrations are made to characterize the physical behavior of different parameters and significance of such system parameters over the momentum, concentration and energy profiles. Moreover, to validate our current results, some limiting models such as fractional and classical fluid models for Maxwell and Newtonian are recovered, in the presence of with/without slip boundary wall conditions. Further, it is observed from the graphs the velocity curves for classical fluid models are relatively higher than fractional fluid models. A comparative analysis between fractional and classical models depicts that the Prabhakar fractional model explains the memory effects more adequately. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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6 pages, 256 KiB

Open AccessArticle

A Septuple Integral of the Product of Three Bessel Functions of the First Kind J_α(tβ)J_γ(xδ)J_η(yθ): Derivation and Evaluation over General Indices

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(4), 730; https://doi.org/10.3390/sym14040730 - 03 Apr 2022

Viewed by 1171

Abstract

Closed expressions for a number of septuple integrals involving the product of three Bessel functions of the first kind

J_{α} (t β) J_{γ} (x δ) J_{η} (y θ)

when the orders [...] Read more.

Closed expressions for a number of septuple integrals involving the product of three Bessel functions of the first kind

J_{α} (t β) J_{γ} (x δ) J_{η} (y θ)

when the orders

α, γ, η

are large, are derived in terms of the Hurwitz–Lerch zeta function

Φ (z, s, v)

. The integrals are not easy to to evaluate for complex values of the parameters. All the results in this work are new. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

13 pages, 928 KiB

Open AccessArticle

Fast Overlapping Block Processing Algorithm for Feature Extraction

by Sadiq H. Abdulhussain, Basheera M. Mahmmod, Jan Flusser, Khaled A. AL-Utaibi and Sadiq M. Sait

Symmetry 2022, 14(4), 715; https://doi.org/10.3390/sym14040715 - 01 Apr 2022

Cited by 14 | Viewed by 1873

Abstract

In many video and image processing applications, the frames are partitioned into blocks, which are extracted and processed sequentially. In this paper, we propose a fast algorithm for calculation of features of overlapping image blocks. We assume the features are projections of the [...] Read more.

In many video and image processing applications, the frames are partitioned into blocks, which are extracted and processed sequentially. In this paper, we propose a fast algorithm for calculation of features of overlapping image blocks. We assume the features are projections of the block on separable 2D basis functions (usually orthogonal polynomials) where we benefit from the symmetry with respect to spatial variables. The main idea is based on a construction of auxiliary matrices that virtually extends the original image and makes it possible to avoid a time-consuming computation in loops. These matrices can be pre-calculated, stored and used repeatedly since they are independent of the image itself. We validated experimentally that the speed up of the proposed method compared with traditional approaches approximately reaches up to 20 times depending on the block parameters. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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11 pages, 777 KiB

Open AccessArticle

Novel Oscillation Theorems and Symmetric Properties of Nonlinear Delay Differential Equations of Fourth-Order with a Middle Term

by Barakah Almarri, S. Janaki, V. Ganesan, Ali Hasan Ali, Kamsing Nonlaopon and Omar Bazighifan

Symmetry 2022, 14(3), 585; https://doi.org/10.3390/sym14030585 - 16 Mar 2022

Cited by 22 | Viewed by 1688

Abstract

The goal of this paper was to study the oscillations of a class of fourth-order nonlinear delay differential equations with a middle term. Novel oscillation theorems built on a proper Riccati-type transformation, the comparison approach, and integral-averaging conditions were developed, and several symmetric [...] Read more.

The goal of this paper was to study the oscillations of a class of fourth-order nonlinear delay differential equations with a middle term. Novel oscillation theorems built on a proper Riccati-type transformation, the comparison approach, and integral-averaging conditions were developed, and several symmetric properties of the solutions are presented. For the validation of these theorems, several examples are given to highlight the core results. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

8 pages, 303 KiB

Open AccessArticle

Symmetric and Non-Oscillatory Characteristics of the Neutral Differential Equations Solutions Related to p-Laplacian Operators

by Barakah Almarri, Ali Hasan Ali, Khalil S. Al-Ghafri, Alanoud Almutairi, Omar Bazighifan and Jan Awrejcewicz

Symmetry 2022, 14(3), 566; https://doi.org/10.3390/sym14030566 - 13 Mar 2022

Cited by 24 | Viewed by 1682

Abstract

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these [...] Read more.

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these equations, and we showed their symmetric and non-oscillatory characteristics. These results complement some previously published articles. To find out the effectiveness of these results and validate the proposed work, two examples were discussed at the end of the paper. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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12 pages, 290 KiB

Open AccessArticle

On (p, q)-Sine and (p, q)-Cosine Fubini Polynomials

by Waseem Ahmad Khan, Ghulam Muhiuddin, Ugur Duran and Deena Al-Kadi

Symmetry 2022, 14(3), 527; https://doi.org/10.3390/sym14030527 - 04 Mar 2022

Cited by 5 | Viewed by 1644

Abstract

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important [...] Read more.

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important that any polynomial have explicit formulas, symmetric identities, summation formulas, and relations with other polynomials. In this work, the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are introduced and multifarious abovementioned properties for these polynomials are derived by utilizing some series manipulation methods.

(p, q)

-derivative operator rules and

(p, q)

-integral representations for the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are also given. Moreover, several correlations related to both the

(p, q)

-Bernoulli, Euler, and Genocchi polynomials and the

(p, q)

-Stirling numbers of the second kind are developed. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

15 pages, 1545 KiB

Open AccessArticle

Stable Calculation of Discrete Hahn Functions

by Albertus C. den Brinker

Symmetry 2022, 14(3), 437; https://doi.org/10.3390/sym14030437 - 23 Feb 2022

Viewed by 1134

Abstract

Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed [...] Read more.

Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in

ℓ_{2}

derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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59 pages, 656 KiB

Open AccessArticle

Product of Hessians and Discriminant of Critical Points of Level Function Attached to Sphere Arrangement

by Kazuhiko Aomoto and Masahiko Ito

Symmetry 2022, 14(2), 374; https://doi.org/10.3390/sym14020374 - 13 Feb 2022

Viewed by 1361

Abstract

We state the product formulae of the values of the levels of functions at critical points involved in asymptotic behaviors of hypergeometric integrals associated with symmetric arrangements of three-dimensional spheres. We show, in an explicit way, how the product of the Hessian, regarding [...] Read more.

We state the product formulae of the values of the levels of functions at critical points involved in asymptotic behaviors of hypergeometric integrals associated with symmetric arrangements of three-dimensional spheres. We show, in an explicit way, how the product of the Hessian, regarding the level functions at all critical points, is related to the behavior of its critical points. We also state two conjectures concerning the same problem associated with general hypersphere arrangements. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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7 pages, 265 KiB

Open AccessArticle

j-Dimensional Integral Involving the Logarithmic and Exponential Functions: Derivation and Evaluation

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(2), 280; https://doi.org/10.3390/sym14020280 - 29 Jan 2022

Viewed by 1617

Abstract

In the fields of science and engineering, tasks involving repeated integrals appear on occasion. The authors’ study on repeated integrals of a class of exponential and logarithmic functions is presented in this publication. The paper includes several examples that demonstrate the evaluation of [...] Read more.

In the fields of science and engineering, tasks involving repeated integrals appear on occasion. The authors’ study on repeated integrals of a class of exponential and logarithmic functions is presented in this publication. The paper includes several examples that demonstrate the evaluation of the analytical parts of the multi-dimensional integral derived. All the results in this work are new. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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12 pages, 322 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus

by Zine El Abiddine Fellah, Mohamed Fellah, Rémi Roncen, Nicholas O. Ongwen, Erick Ogam and Claude Depollier

Symmetry 2022, 14(2), 233; https://doi.org/10.3390/sym14020233 - 25 Jan 2022

Cited by 7 | Viewed by 2358

Abstract

A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case [...] Read more.

A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot’s theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The basic solution of the wave equation is obtained in the time domain by analytically calculating Green’s function of the porous medium and using the properties of the Laplace transforms. Fractional derivatives are used to describe, in the time domain, the fluid–structure interactions, which are of the inertial, viscous, and thermal kind. The solution to the fractional-order wave equation represents the radiation field in the porous medium emitted by a point source. An important result obtained in this study is that the solution of the fractional equation is expressed by recurrence relations that are the consequence of the modified Bessel function of the third kind, which represents a physical solution of the wave equation. This theoretical work with analytical results opens up prospects for the resolution of forward and inverse problems allowing the characterization of a porous medium using spherical waves. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

17 pages, 3480 KiB

Open AccessArticle

Double Diffusive Magneto-Free-Convection Flow of Oldroyd-B Fluid over a Vertical Plate with Heat and Mass Flux

by Muhammad Bilal Riaz, Aziz Ur Rehman, Jan Awrejcewicz and Fahd Jarad

Symmetry 2022, 14(2), 209; https://doi.org/10.3390/sym14020209 - 21 Jan 2022

Cited by 16 | Viewed by 2057

Abstract

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in an Oldroyd-B fluid flow based on the fundamental symmetry that are presented in non-dimensional form and are applied to a moving heated vertical plate as the [...] Read more.

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in an Oldroyd-B fluid flow based on the fundamental symmetry that are presented in non-dimensional form and are applied to a moving heated vertical plate as the boundary layer flow up, with the existence of an external magnetic field that is either moving or fixed consistent with the plate. The thermal transport phenomenon in the presence of constant concentration, coupled with a first order chemical reaction under the exponential heating of the symmetry of fluid flow, is analyzed. The Laplace transform method is applied symmetrically to tackle the non-dimensional partial differential equations for velocity, mass and energy. The contribution of mass, thermal and mechanical components on the dynamics of fluid are presented and discussed independently. An interesting property regarding the behavior of the fluid velocity is found when the movement is observed in the magnetic intensity along with the plate. In that situation, the fluid velocity is not zero when it is far and away from the plate. Moreover, the heat transfer aspects, flow dynamics and their credence on the parameters are drawn out by graphical illustrations. Furthermore, some special cases for the movement of the plate are also studied. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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6 pages, 254 KiB

Open AccessArticle

A Quadruple Integral Containing the Gegenbauer Polynomial C_n^(λ)(x): Derivation and Evaluation

by Robert Reynolds and Allan Stauffer

Symmetry 2022, 14(2), 205; https://doi.org/10.3390/sym14020205 - 21 Jan 2022

Viewed by 1678

Abstract

A four-dimensional integral containing

g (x, y, z, t) C_{n}^{(λ)} (x)

is derived.

C_{n}^{(λ)} (x)

is the Gegenbauer polynomial, [...] Read more.

A four-dimensional integral containing

g (x, y, z, t) C_{n}^{(λ)} (x)

is derived.

C_{n}^{(λ)} (x)

is the Gegenbauer polynomial,

g (x, y, z, t)

is a product of the generalized logarithm quotient functions and the integral is taken over the region

0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1, 0 \leq t \leq 1

. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

15 pages, 292 KiB

Open AccessArticle

Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

by Cristina B. Corcino, Baby Ann A. Damgo, Joy Ann A. Cañete and Roberto B. Corcino

Symmetry 2022, 14(1), 53; https://doi.org/10.3390/sym14010053 - 01 Jan 2022

Cited by 2 | Viewed by 1293

Abstract

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter

λ

. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases

λ = 1

and [...] Read more.

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter

λ

. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases

λ = 1

and

λ = - 1

were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

10 pages, 280 KiB

Open AccessArticle

Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

by Cristina B. Corcino, Roberto B. Corcino, Baby Ann A. Damgo and Joy Ann A. Cañete

Symmetry 2022, 14(1), 35; https://doi.org/10.3390/sym14010035 - 28 Dec 2021

Cited by 6 | Viewed by 1407

Abstract

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using [...] Read more.

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

15 pages, 300 KiB

Open AccessArticle

A Subclass of Janowski Starlike Functions Involving Mathieu-Type Series

by Dong Liu, Serkan Araci and Bilal Khan

Symmetry 2022, 14(1), 2; https://doi.org/10.3390/sym14010002 - 21 Dec 2021

Viewed by 2210

Abstract

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new [...] Read more.

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

20 pages, 817 KiB

Open AccessArticle

Some New Reverse Hilbert’s Inequalities on Time Scales

by Ghada AlNemer, Ahmed I. Saied, Mohammed Zakarya, Hoda A. Abd El-Hamid, Omar Bazighifan and Haytham M. Rezk

Symmetry 2021, 13(12), 2431; https://doi.org/10.3390/sym13122431 - 15 Dec 2021

Cited by 12 | Viewed by 1869

Abstract

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for [...] Read more.

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

21 pages, 333 KiB

Open AccessArticle

New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

by Waleed Mohamed Abd-Elhameed and Seraj Omar Alkhamisi

Symmetry 2021, 13(12), 2407; https://doi.org/10.3390/sym13122407 - 13 Dec 2021

Cited by 14 | Viewed by 2093

Abstract

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials [...] Read more.

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

17 pages, 303 KiB

Open AccessArticle

On Lommel Matrix Polynomials

by Ayman Shehata

Symmetry 2021, 13(12), 2335; https://doi.org/10.3390/sym13122335 - 06 Dec 2021

Cited by 2 | Viewed by 1593

Abstract

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and [...] Read more.

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and integral representations for Lommel matrix polynomials and discuss its various special cases. Finally, we establish an entire function, order, type, explicit representation and several properties of modified Lommel matrix polynomials. There are also several unique examples of our comprehensive results constructed. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

10 pages, 266 KiB

Open AccessArticle

Integrable Nonlocal PT-Symmetric Modified Korteweg-de Vries Equations Associated with so(3, \({\mathbb{R}}\))

by Wen-Xiu Ma

Symmetry 2021, 13(11), 2205; https://doi.org/10.3390/sym13112205 - 19 Nov 2021

Cited by 9 | Viewed by 1329

Abstract

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra

so (3, R)

. The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal complex reverse-spacetime and real reverse-spacetime modified Korteweg-de Vries equations [...] Read more.

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra

so (3, R)

. The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal complex reverse-spacetime and real reverse-spacetime modified Korteweg-de Vries equations associated with

so (3, R)

. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

16 pages, 330 KiB

Open AccessArticle

Existence Results of a Nonlocal Fractional Symmetric Hahn Integrodifference Boundary Value Problem

by Rujira Ouncharoen, Nichaphat Patanarapeelert and Thanin Sitthiwirattham

Symmetry 2021, 13(11), 2174; https://doi.org/10.3390/sym13112174 - 12 Nov 2021

Viewed by 971

Abstract

The existence of solutions of nonlocal fractional symmetric Hahn integrodifference boundary value problem is studied. We propose a problem of five fractional symmetric Hahn difference operators and three fractional symmetric Hahn integrals of different orders. We first convert our nonlinear problem into a [...] Read more.

The existence of solutions of nonlocal fractional symmetric Hahn integrodifference boundary value problem is studied. We propose a problem of five fractional symmetric Hahn difference operators and three fractional symmetric Hahn integrals of different orders. We first convert our nonlinear problem into a fixed point problem by considering a linear variant of the problem. When the fixed point operator is available, Banach and Schauder’s fixed point theorems are used to prove the existence results of our problem. Some properties of

(q, ω)

-integral are also presented in this paper as a tool for our calculations. Finally, an example is also constructed to illustrate the main results. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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