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

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 2548

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Special Issue Editors

Prof. Dr. Waleed Mohamed Abd-Elhameed

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Guest Editor

1. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
2. Department of Mathematics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
Interests: special functions; numerical analysis; sequences of polynomials and numbers; boundary value problems fractional differential equations

Dr. Youssri Youssri

E-Mail Website
Guest Editor

Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Interests: numerical analysis; special functions; fractional differential equations
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Anna Napoli

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Guest Editor

Department of Mathematics and Computer Science, University of Calabria, Via Pietro Bucci, Cubo 30/A, 87036 Rende, CS, Italy
Interests: polynomials and their applications in approximation theory; boundary value problems; numerical quadrature; zeros of functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special functions, including orthogonal polynomials with symmetric and nonsymmetric parameters, are important in many areas of mathematics and in the applied sciences. For example, they are very important in the numerical analysis of different types of differential equations.

An orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are classical orthogonal polynomials, consisting of Hermite polynomials, Laguerre polynomials, and Jacobi polynomials, and their special cases for symmetric parameters, namely, Gegenbauer polynomials, which form the most important class of Jacobi polynomials; other special cases include Chebyshev polynomials and Legendre polynomials.

P.L. Chebyshev's work on continued fractions in the late 19th century inspired further research into orthogonal polynomials by A. A. Markov and T. J. Stieltjes. Many other areas of mathematics and science also use them, such as numerical analysis (quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of random matrices, integrable systems, etc.), and number theory.

This Special Issue welcomes papers devoted to the theory and applications of special functions, including symmetric and non-symmetric orthogonal polynomials. Emphasis will be placed on the use of any applied polynomial set to handle various differential and integral problems. Both papers concerned with their applicability in many disciplines, and papers on other topics in mathematics and the applied sciences where polynomials play an essential role are cordially invited. Articles of a survey nature will also be considered.

Prof. Dr. Waleed Mohamed Abd-Elhameed
Dr. Youssri Youssri
Prof. Dr. Anna Napoli
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

orthogonal polynomials
hypergeometric functions
special functions
sequences of polynomials
linearization problems
connection formulas between symmetric and nonsymmetric polynomials
differential equations
integral equations
spectral methods
quadrature theory

Published Papers (4 papers)

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Research

Open AccessArticle

Special Functions and Its Application in Solving Two Dimensional Hyperbolic Partial Differential Equation of Telegraph Type

Ishtiaq Ali

Maliha Tehseen Saleem

and

Azhar ul Din

Symmetry 2023, 15(4), 847; https://doi.org/10.3390/sym15040847 - 02 Apr 2023

Viewed by 526

Abstract

In this article, we use the applications of special functions in the form of Chebyshev polynomials to find the approximate solution of hyperbolic partial differential equations (PDEs) arising in the mathematical modeling of transmission line subject to appropriate symmetric Dirichlet and Neumann boundary conditions. The special part of the model equation is discretized using a Chebyshev differentiation matrix, which is centro-asymmetric using the symmetric collocation points as grid points, while the time derivative is discretized using the standard central finite difference scheme. One of the disadvantages of the Chebyshev differentiation matrix is that the resultant matrix, which is obtained after replacing the special coordinates with the derivative of Chebyshev polynomials, is dense and, therefore, needs more computational time to evaluate the resultant algebraic equation. To overcome this difficulty, an algorithm consisting of fast Fourier transformation is used. The main advantage of this transformation is that it significantly reduces the computational cost needed for N collocation points. It is shown that the proposed scheme converges exponentially, provided the data are smooth in the given equations. A number of numerical experiments are performed for different time steps and compared with the analytical solution, which further validates the accuracy of our proposed scheme. Full article

(This article belongs to the Special Issue New Advances in Special Functions and Their Applications in Science and Mathematics)

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Open AccessArticle

Certain Properties and Applications of Convoluted Δ_h Multi-Variate Hermite and Appell Sequences

Shahid Ahmad Wani

Ibtehal Alazman

and

Badr Saad T. Alkahtani

Symmetry 2023, 15(4), 828; https://doi.org/10.3390/sym15040828 - 29 Mar 2023

Viewed by 545

Abstract

This study follows the line of research that by employing the monomiality principle, new outcomes are produced. Thus, in line with prior facts, our aim is to introduce the

Δ_{h}

multi-variate Hermite Appell polynomials [...] Read more.

This study follows the line of research that by employing the monomiality principle, new outcomes are produced. Thus, in line with prior facts, our aim is to introduce the

Δ_{h}

multi-variate Hermite Appell polynomials

_{Δ_{h}}_{H} A_{m}^{[r]} (q_{1}, q_{2}, \dots, q_{r}; h)

. Further, we obtain their recurrence sort of relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families and thus operational principle satisfied by these polynomials is derived and will prove beneficial for future observations. Further, a few members of the

Δ_{h}

Appell polynomial family are considered and their corresponding results are derived accordingly. Full article

(This article belongs to the Special Issue New Advances in Special Functions and Their Applications in Science and Mathematics)

Open AccessArticle

Spectral Solutions of Even-Order BVPs Based on New Operational Matrix of Derivatives of Generalized Jacobi Polynomials

Waleed Mohamed Abd-Elhameed

Badah Mohamed Badah

Amr Kamel Amin

and

Muhammad Mahmoud Alsuyuti

Symmetry 2023, 15(2), 345; https://doi.org/10.3390/sym15020345 - 26 Jan 2023

Cited by 1 | Viewed by 512

Abstract

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given. Full article

(This article belongs to the Special Issue New Advances in Special Functions and Their Applications in Science and Mathematics)

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Open AccessArticle

Some Novel Formulas of Lucas Polynomials via Different Approaches

Waleed Mohamed Abd-Elhameed

and

Anna Napoli

Symmetry 2023, 15(1), 185; https://doi.org/10.3390/sym15010185 - 08 Jan 2023

Cited by 2 | Viewed by 590

Abstract

Some new formulas related to the well-known symmetric Lucas polynomials are the primary focus of this article. Different approaches are used for establishing these formulas. A matrix approach to Lucas polynomials is followed in order to obtain some fundamental properties. Particularly, some recurrence relations and determinant forms are determined by suitable Hessenberg matrices. Conjugate Lucas polynomials and generating functions are derived and examined. Several connection problems between the Lucas polynomials and other celebrated symmetric and non-symmetric orthogonal polynomials such as the first and second kinds of Chebyshev polynomials and their shifted counterparts are solved. We prove that several argument-type hypergeometric functions are involved in the connection coefficients. In addition, we construct new formulas for high-order derivatives of Lucas polynomials in terms of their original polynomials, as well as formulas for repeated integrals of Lucas polynomials. Full article

(This article belongs to the Special Issue New Advances in Special Functions and Their Applications in Science and Mathematics)

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