Theory and Applications of Special Functions in Mathematical Physics

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

Deadline for manuscript submissions: closed (15 January 2022) | Viewed by 17163

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Special Issue Information

Dear Colleagues,

Different applications of modern engineering and physical sciences require thorough knowledge of applied mathematics, particularly special functions. These are frequently adopted in acoustics, thermodynamics, electromagnetics, and optics, to express the approximate or exact analytical solution of complex problems, thus providing a better understanding of and meaningful insight into underlying properties and mechanisms.

In this Special Issue, we focus on the application of classical and higher-order special functions to advanced problems of mathematical physics that are characterized by specific (i.e., rectangular, cylindrical, and spherical) symmetry or, conversely, rely on more unconventional models. Attention is given, also, to the illustration of properties of novel special functions, with particular attention paid to the relevant governing differential equation; recurrence formulae; as well as efficient computational algorithms, such as those based on uniform asymptotic representations for small and large arguments.

We look forward to your contributions to review- and original-research articles dealing with the recent advances in the theory and applications of special functions.

Prof. Dr. Diego Caratelli
Guest Editor

Manuscript Submission Information

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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.

Published Papers (9 papers)

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Research

11 pages, 284 KiB  
Article
Certain Finite Integrals Related to the Products of Special Functions
by Dinesh Kumar, Frédéric Ayant, Suphawat Asawasamrit and Jessada Tariboon
Symmetry 2021, 13(11), 2013; https://doi.org/10.3390/sym13112013 - 23 Oct 2021
Cited by 1 | Viewed by 1397
Abstract
The aim of this paper is to establish a theorem associated with the product of the Aleph-function, the multivariable Aleph-function, and the general class of polynomials. The results of this theorem are unified in nature and provide a very large number of analogous [...] Read more.
The aim of this paper is to establish a theorem associated with the product of the Aleph-function, the multivariable Aleph-function, and the general class of polynomials. The results of this theorem are unified in nature and provide a very large number of analogous results (new or known) involving simpler special functions and polynomials (of one or several variables) as special cases. The derived results lead to significant applications in physics and engineering sciences. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
18 pages, 316 KiB  
Article
Applications of Generalized q-Difference Equations for General q-Polynomials
by Zeya Jia, Bilal Khan, Qiuxia Hu and Dawei Niu
Symmetry 2021, 13(7), 1222; https://doi.org/10.3390/sym13071222 - 7 Jul 2021
Cited by 8 | Viewed by 1576
Abstract
Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(,1,1,q) is the [...] Read more.
Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(,1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
35 pages, 904 KiB  
Article
Fourier Transforms of Some Finite Bivariate Orthogonal Polynomials
by Esra Güldoğan Lekesiz, Rabia Aktaş and Mohammad Masjed-Jamei
Symmetry 2021, 13(3), 452; https://doi.org/10.3390/sym13030452 - 10 Mar 2021
Cited by 2 | Viewed by 1282
Abstract
In this paper, we first obtain the Fourier transforms of some finite bivariate orthogonal polynomials and then by using the Parseval identity, we introduce some new families of bivariate orthogonal functions. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
14 pages, 1863 KiB  
Article
Generalized Bessel Functions and Their Use in Bremsstrahlung and Multi-Photon Processes
by Giuseppe Dattoli, Emanuele Di Palma, Silvia Licciardi and Elio Sabia
Symmetry 2021, 13(2), 159; https://doi.org/10.3390/sym13020159 - 20 Jan 2021
Cited by 3 | Viewed by 2253
Abstract
The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed [...] Read more.
The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
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14 pages, 20132 KiB  
Article
Lacunary Möbius Fractals on the Unit Disk
by L. K. Mork, Keith Sullivan and Darin J. Ulness
Symmetry 2021, 13(1), 91; https://doi.org/10.3390/sym13010091 - 6 Jan 2021
Cited by 2 | Viewed by 1839
Abstract
Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk [...] Read more.
Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the Möbius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
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16 pages, 344 KiB  
Article
A Note on Generalized q-Difference Equations and Their Applications Involving q-Hypergeometric Functions
by Hari M. Srivastava, Jian Cao and Sama Arjika
Symmetry 2020, 12(11), 1816; https://doi.org/10.3390/sym12111816 - 2 Nov 2020
Cited by 19 | Viewed by 1919
Abstract
Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in [...] Read more.
Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators T(a,b,c,d,e,yDx) and E(a,b,c,d,e,yθx) to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
19 pages, 293 KiB  
Article
More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral
by M. Zakarya, H. A. Abd El-Hamid, Ghada AlNemer and H. M. Rezk
Symmetry 2020, 12(10), 1716; https://doi.org/10.3390/sym12101716 - 18 Oct 2020
Cited by 8 | Viewed by 1612
Abstract
In this paper, we investigate some new generalizations and refinements for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various [...] Read more.
In this paper, we investigate some new generalizations and refinements for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi’s inequalities achieved on time scales and also include some integral disparities as particular cases when T=R. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
12 pages, 2845 KiB  
Article
Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship
by Weidong Yang, Xuehui Chen, Yuan Meng, Xinru Zhang and Shiyun Mi
Symmetry 2020, 12(9), 1446; https://doi.org/10.3390/sym12091446 - 2 Sep 2020
Cited by 2 | Viewed by 1941
Abstract
In this paper, we develop a new time-space fractional constitution relation to study the unsteady boundary layer flow over a stretching sheet. For the convenience of calculation, the boundary layer flow is simulated as a symmetrical rectangular area. The implicit difference method combined [...] Read more.
In this paper, we develop a new time-space fractional constitution relation to study the unsteady boundary layer flow over a stretching sheet. For the convenience of calculation, the boundary layer flow is simulated as a symmetrical rectangular area. The implicit difference method combined with an L1-algorithm and shift Grünwald scheme is used to obtain the numerical solutions of the fractional governing equation. The validity and solvability of the present numerical method are analyzed systematically. The numerical results show that the thickness of the velocity boundary layer increases with an increase in the space fractional parameter γ. For a different stress fractional parameter α, the viscoelastic fluid will exhibit viscous or elastic behavior, respectively. Furthermore, the numerical method in this study is validated and can be extended to other time-space fractional boundary layer models. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
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7 pages, 478 KiB  
Article
A Note on the Orthogonality Properties of the Pseudo-Chebyshev Functions
by Diego Caratelli and Paolo Emilio Ricci
Symmetry 2020, 12(8), 1273; https://doi.org/10.3390/sym12081273 - 2 Aug 2020
Cited by 3 | Viewed by 1986
Abstract
A novel class of pseudo-Chebyshev functions has been recently introduced, and the relevant analytical properties in terms of governing differential equation, recurrence formulae, and orthogonality have been analyzed in detail for half-integer degrees. In this paper, the previous studies are extended to the [...] Read more.
A novel class of pseudo-Chebyshev functions has been recently introduced, and the relevant analytical properties in terms of governing differential equation, recurrence formulae, and orthogonality have been analyzed in detail for half-integer degrees. In this paper, the previous studies are extended to the general case of rational degree. In particular, it is shown that the orthogonality properties of the pseudo-Chebyshev functions do not hold any longer. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)
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