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Symmetry
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Research in Special Functions
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Research in Special Functions

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

Deadline for manuscript submissions: 28 February 2025 | Viewed by 5400

Special Issue Editors

Prof. Dr. Ilya Shilin

E-Mail Website
Guest Editor
1. Department of Higher Mathematics, National Research University MPEI, Moscow 111250, Russia
2. Department of Algebra, Moscow State Pedagogical University, Moscow 119991, Russia
Interests: special functions of mathematical physics; group theoretical approach to special functions; integral transforms
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Junesang Choi

E-Mail Website
Guest Editor
Department of Mathematics, Dongguk University, Wise Campus, Gyeongju 38066, Republic of Korea
Interests: special functions; analytic number theory; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to present novel results for special functions arising in various areas of contemporary mathematics, including mathematical physics, theory of ODE and PDE, number theory, discrete mathematics, harmonic analysis, theory of integral transforms, Lie groups and Lie algebras representation theory, q-calculus, fractional calculus, etc. We expect that this Special Issue will address both classical special functions and their numerous extensions, including q-analogues, fractional analogues, and hyper (multi-index) analogues.

We look forward to receiving your contributions including new properties, integrals, series and recurrent relations, formulas for asymptotic behavior and values of integral operators, new results for analytic continuations, etc. We invite authors to present new theorems describing connections between special functions of mathematical physics and their q-analogues with classical Lie groups and algebras and quantum groups and Hopf algebras, respectively.

We are also interested in various applications of special functions, since “A function is a special function if it occurs often enough so that it gets a name” (Richard Askey).

You may choose our Joint Special Issue in Axioms [ISSN 2075-1680, SCIE Indexed, IF: 1.824].

Prof. Dr. Ilya Shilin
Prof. Dr. Junesang Choi
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.

Keywords

  • special functions
  • hyperfunction
  • q-special functions
  • special functions of fractional calculus
  • group theoretical approach to special functions
  • harmonic analysis
  • special functions of number theory
  • generalized hypergeometric function
  • special functions of several variables
  • generalized fractional integrals and fractional derivatives

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (5 papers)

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Research

15 pages, 313 KiB  
Article
On a Class of Generalized Multivariate Hermite–Humbert Polynomials via Generalized Fibonacci Polynomials
by Noor Alam, Shahid Ahmad Wani, Waseem Ahmad Khan, Ketan Kotecha, Hasan Nihal Zaidi, Fakhredine Gassem and Anas Altaleb
Symmetry 2024, 16(11), 1415; https://doi.org/10.3390/sym16111415 - 23 Oct 2024
Viewed by 542
Abstract
This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems [...] Read more.
This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields. Full article
(This article belongs to the Special Issue Research in Special Functions)
12 pages, 267 KiB  
Article
Linear Arrangement of Euler Sums with Multiple Argument
by Anthony Sofo
Symmetry 2024, 16(10), 1322; https://doi.org/10.3390/sym16101322 - 8 Oct 2024
Viewed by 777
Abstract
We investigate the linear arrangement of Euler harmonic sums that may be expressed in closed form in terms of special functions such as the classical Riemann zeta function and the Dirichlet eta function. Particular emphasis is given to Euler harmonic sums with even [...] Read more.
We investigate the linear arrangement of Euler harmonic sums that may be expressed in closed form in terms of special functions such as the classical Riemann zeta function and the Dirichlet eta function. Particular emphasis is given to Euler harmonic sums with even weight. New examples highlighting the theorems will be presented. Full article
(This article belongs to the Special Issue Research in Special Functions)
14 pages, 287 KiB  
Article
On Fractional Operators Involving the Incomplete Mittag-Leffler Matrix Function and Its Applications
by Ahmed Bakhet, Shahid Hussain and Mohra Zayed
Symmetry 2024, 16(8), 963; https://doi.org/10.3390/sym16080963 - 29 Jul 2024
Viewed by 986
Abstract
In this study, we derive multiple incomplete matrix Mittag-Leffler (ML) functions. We systematically investigate several properties of these incomplete matrix ML functions, which include some general properties and distinct representations of integral transforms. We further study the properties of the Riemann–Liouville fractional integrals [...] Read more.
In this study, we derive multiple incomplete matrix Mittag-Leffler (ML) functions. We systematically investigate several properties of these incomplete matrix ML functions, which include some general properties and distinct representations of integral transforms. We further study the properties of the Riemann–Liouville fractional integrals and derivatives related to the incomplete matrix ML functions. Additionally, some interesting special cases of this work are highlighted. Finally, we establish the solution to the kinetic equations as an application. Full article
(This article belongs to the Special Issue Research in Special Functions)
16 pages, 295 KiB  
Article
Convolution Theorem for (p,q)-Gamma Integral Transforms and Their Application to Some Special Functions
by Shrideh Al-Omari, Wael Salameh and Hamzeh Zureigat
Symmetry 2024, 16(7), 882; https://doi.org/10.3390/sym16070882 - 11 Jul 2024
Viewed by 933
Abstract
This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning [...] Read more.
This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems. Full article
(This article belongs to the Special Issue Research in Special Functions)
23 pages, 419 KiB  
Article
Extended Exton’s Triple and Horn’s Double Hypergeometric Functions and Associated Bounding Inequalities
by Rakesh Kumar Parmar, Junesang Choi and Saravanan S.
Symmetry 2023, 15(6), 1132; https://doi.org/10.3390/sym15061132 - 23 May 2023
Cited by 1 | Viewed by 1445
Abstract
This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. [...] Read more.
This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H4 and X8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X8,p and several bounding inequalities for H4,p and X8,p. We explore the utilization of the H4,p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry. Full article
(This article belongs to the Special Issue Research in Special Functions)
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