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Symmetry
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Special Functions and Orthogonal Polynomials: Symmetry and Applications
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Special Functions and Orthogonal Polynomials: Symmetry and Applications

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

Deadline for manuscript submissions: closed (15 June 2023) | Viewed by 4582

Special Issue Editor

Dr. Dmitrii Karp

E-Mail Website
Guest Editor
Departement of Mathematics, Holon Institute of Technology, Holon 5810201, Israel
Interests: special functions; function theory; inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special functions is an unfortunate denotation for numerous concrete functions playing an important role in various branches of mathematics and applied sciences. Paul Turán is accredited with the authorship of suggestion to call them “useful functions”. They appear in a variety of ways, such as solutions of differential/integral equations, densities of probability distributions, counting combinatorial objects and number-theoretic quantities, evaluation of integrals and summation of series, solutions of the Riemann–Hilbert problems, explicit expression for (multiple) orthogonal polynomials, fractional calculus, and matrix elements of group representations, to name a few.  

This Special Issue welcomes papers devoted to both the theory and applications of special functions. Emphasis will be placed on hypergeometric and related functions and orthogonal polynomials. The fundamental theory of these functions, such as representations, transformation and summation formulas, asymptotic approximations, inequalities, zeros, monotonicity, and complex analytic properties will be at the focus. The second main theme is orthogonal and multiple orthogonal polynomials; both papers concerned with their concrete families and with general properties are welcome. Furthermore, papers on other topics in both mathematics and sciences, where special functions play an important role, are also cordially invited. Articles of survey nature will also be considered.

Dr. Dmitrii Karp
Guest Editor

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
  • orthogonal polynomials
  • discrete orthogonal polynomials
  • hypergeometric functions
  • Mellin-Barnes integrals
  • basic hypergeometric functions
  • Heun functions
  • Painlevé transcendent
  • multiple orthogonal polynomials
  • asymptotic
  • inequalities

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Published Papers (3 papers)

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Research

20 pages, 1302 KiB  
Article
Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling
by Noor Alam, Waseem Ahmad Khan, Can Kızılateş, Sofian Obeidat, Cheon Seoung Ryoo and Nabawia Shaban Diab
Symmetry 2023, 15(7), 1358; https://doi.org/10.3390/sym15071358 - 4 Jul 2023
Cited by 5 | Viewed by 1046
Abstract
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving [...] Read more.
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. Full article
(This article belongs to the Special Issue Special Functions and Orthogonal Polynomials: Symmetry and Applications)
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18 pages, 365 KiB  
Article
Symmetry Analysis of a Model of Option Pricing and Hedging
by Sergey M. Sitnik, Khristofor V. Yadrikhinskiy and Vladimir E. Fedorov
Symmetry 2022, 14(9), 1841; https://doi.org/10.3390/sym14091841 - 5 Sep 2022
Cited by 7 | Viewed by 1400
Abstract
The Guéant and Pu model of option pricing and hedging, which takes into account transaction costs, and the impact of operations on the market is studied by group analysis methods. The infinite-dimensional continuous group of equivalence transforms of the model is found. It [...] Read more.
The Guéant and Pu model of option pricing and hedging, which takes into account transaction costs, and the impact of operations on the market is studied by group analysis methods. The infinite-dimensional continuous group of equivalence transforms of the model is found. It is applied to get the group classification of the model under consideration. In addition to the general case, the classification contains three specifications of a free element in the equation, which correspond to models with groups of symmetries of a special kind. Optimal systems of subalgebras for some concrete models from the obtained classification are derived and used for the calculation of according invariant submodels. Full article
(This article belongs to the Special Issue Special Functions and Orthogonal Polynomials: Symmetry and Applications)
31 pages, 462 KiB  
Article
Beyond the Beta Integral Method: Transformation Formulas for Hypergeometric Functions via Meijer’s G Function
by Dmitrii Karp and Elena Prilepkina
Symmetry 2022, 14(8), 1541; https://doi.org/10.3390/sym14081541 - 27 Jul 2022
Cited by 4 | Viewed by 1459
Abstract
The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Meijer’s G function. By the [...] Read more.
The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Meijer’s G function. By the application of our method to known transformation formulas, we derive about forty hypergeometric identities, the majority of which are believed to be new. Full article
(This article belongs to the Special Issue Special Functions and Orthogonal Polynomials: Symmetry and Applications)
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