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Symmetry
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Special Functions, Integral Transforms and Polynomial Sequences in Real World...
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Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry

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

Deadline for manuscript submissions: 30 November 2024 | Viewed by 6325

Special Issue Editors

Prof. Dr. Pierpaolo Natalini

E-Mail Website
Guest Editor
Department of Mathematics and Physics, Roma Tre University, Rome, Italy
Interests: special functions
Prof. Dr. Sandra Pinelas

E-Mail Website
Co-Guest Editor
Departamento De Ciencias Exatas E Engenharia Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal
Interests: differential equations; difference equations; oscillatory behavior; asymptotic behavior
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The application of special function theory continues to grow in contributions in all areas of the sciences, with particular reference to those of mathematical physics, chemistry, and the biological sciences.

By means of special functions, it is often possible to find explicit solutions of certain ordinary or partial differential equations for particular boundary conditions and sometimes to derive even the best possible numerical approximations. This is borne out of the extensive literature, which has found its way into prestigious dedicated international journals, such as Integral Transforms and Special Functions, Applied Analysis and Discrete Mathematics, and those that refer to fractional calculus, such as Fractional Calculus and Applied Analysis and the International Journal of Applied Mathematics, among others.

The use of computers, which was initially aimed at constructing approximate solutions of analytical problems by means of numerical tables, has also evolved into the possibility of using computer algebra systems such as Mathematica, Maple, etc., which provide solutions in terms of special functions, avoiding the difficulties of discretization problems.

This can often be done by exploiting the particular symmetries of the domains considered, jointly with those of the special functions themselves that are used in analytical solutions. Indeed, symmetry is a powerful means of analyzing and simplifying problems and a useful tool for their solution.

The purpose of this Symmetry Special Issue is to show an overview of current applications of the theory of special functions and transforms in all fields of applied sciences, both in the traditional field of mathematical physics and in that of biological sciences, probability, and mathematical statistics. Contributions on fractional calculus and the analytic theory of numbers are also welcome.

Prof. Dr. Pierpaolo Natalini
Prof. Dr. Sandra Pinelas
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 2400 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.

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Published Papers (6 papers)

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Research

16 pages, 310 KiB  
Article
On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces
by Shrideh Al-Omari and Wael Salameh
Symmetry 2024, 16(10), 1307; https://doi.org/10.3390/sym16101307 - 3 Oct 2024
Viewed by 979
Abstract
In this article, the (p,q)-analogs of the α-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of (p,q)-generalized functions. Two spaces of infinitely [...] Read more.
In this article, the (p,q)-analogs of the α-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of (p,q)-generalized functions. Two spaces of infinitely (p,q)-differentiable functions are defined by introducing two (p,q)-differential symmetric operators. The (p,q)-analogs of the α-th fractional Fourier transform are demonstrated to be continuous and linear between the spaces under discussion. Next, theorems pertaining to specific convolutions are established. This leads to the establishment of multiple symmetric identities, which in turn requires the construction of (p,q)-generalized spaces known as (p,q)-Boehmians. Finally, in addition to deriving the inversion formulas, the generalized (p,q)- analogs of the α-th fractional Fourier transform are introduced, and their general properties are discussed. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
25 pages, 507 KiB  
Article
Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations
by Musa Rahamh GadAllah and Hassan Eltayeb Gadain
Symmetry 2024, 16(9), 1232; https://doi.org/10.3390/sym16091232 - 19 Sep 2024
Viewed by 612
Abstract
In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included [...] Read more.
In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
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16 pages, 274 KiB  
Article
Exploring Properties and Applications of Laguerre Special Polynomials Involving the Δh Form
by Noor Alam, Shahid Ahmad Wani, Waseem Ahmad Khan, Fakhredine Gassem and Anas Altaleb
Symmetry 2024, 16(9), 1154; https://doi.org/10.3390/sym16091154 - 4 Sep 2024
Viewed by 809
Abstract
The primary objective of this research is to introduce and investigate novel polynomial variants termed Δh Laguerre polynomials. This unique polynomial type integrates the monomiality principle alongside operational rules. Through this innovative approach, the study delves into uncharted territory, unveiling fresh insights [...] Read more.
The primary objective of this research is to introduce and investigate novel polynomial variants termed Δh Laguerre polynomials. This unique polynomial type integrates the monomiality principle alongside operational rules. Through this innovative approach, the study delves into uncharted territory, unveiling fresh insights that build upon prior research endeavours. Notably, the Δh Laguerre polynomials exhibit significant utility in the realm of quantum mechanics, particularly in the modelling of entropy within quantum systems. The research meticulously unveils explicit formulas and elucidates the fundamental properties of these polynomials, thereby forging connections with established polynomial categories. By shedding light on the distinct characteristics and functionalities of the Δh Laguerre polynomials, this study contributes significantly to their comprehension and application across diverse mathematical and scientific domains. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
11 pages, 4592 KiB  
Article
Examples of Expansions in Fractional Powers, and Applications
by Diego Caratelli, Pierpaolo Natalini and Paolo Emilio Ricci
Symmetry 2023, 15(9), 1702; https://doi.org/10.3390/sym15091702 - 6 Sep 2023
Cited by 4 | Viewed by 998
Abstract
We approximate the solution of a generalized form of the Bagley–Torvik equation using Taylor’s expansions in fractional powers. Then, we study the fractional Laguerre-type logistic equation by considering the fractional exponential function and its Laguerre-type form. To verify our findings, we conduct numerical [...] Read more.
We approximate the solution of a generalized form of the Bagley–Torvik equation using Taylor’s expansions in fractional powers. Then, we study the fractional Laguerre-type logistic equation by considering the fractional exponential function and its Laguerre-type form. To verify our findings, we conduct numerical tests using the computer algebra program Mathematica©. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
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18 pages, 281 KiB  
Article
On Several Results Associated with the Apéry-like Series
by Prathima Jayarama, Dongkyu Lim and Arjun K. Rathie
Symmetry 2023, 15(5), 1022; https://doi.org/10.3390/sym15051022 - 4 May 2023
Cited by 1 | Viewed by 1097
Abstract
In 1979, Apéry proved the irrationality of ζ(2) and ζ(3). Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite [...] Read more.
In 1979, Apéry proved the irrationality of ζ(2) and ζ(3). Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
11 pages, 285 KiB  
Article
Some Identities with Multi-Generalized q-Hyperharmonic Numbers of Order r
by Zhihua Chen, Neşe Ömür, Sibel Koparal and Waseem Ahmad Khan
Symmetry 2023, 15(4), 917; https://doi.org/10.3390/sym15040917 - 14 Apr 2023
Cited by 2 | Viewed by 1104
Abstract
The main purpose of this paper is to define multiple alternative q-harmonic numbers, Hnk;q and multi-generalized q-hyperharmonic numbers of order r, Hnrk;q by using q-multiple zeta star values (q [...] Read more.
The main purpose of this paper is to define multiple alternative q-harmonic numbers, Hnk;q and multi-generalized q-hyperharmonic numbers of order r, Hnrk;q by using q-multiple zeta star values (q-MZSVs). We obtain some finite sum identities and give some applications of them for certain combinations of q-multiple polylogarithms Liq;k1,k2,…,kdt1,t2,…,td with the help of generating functions. Additionally, one of the applications is the sum involving q-Stirling numbers and q-Bernoulli numbers. Full article
(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)
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