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Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd...
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Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition

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

Deadline for manuscript submissions: 30 September 2025 | Viewed by 3774

Special Issue Editor

Dr. Serkan Araci

E-Mail Website
Guest Editor
Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Türkiye
Interests: q-special functions and q-special polynomials; q-series; analytic number theory; umbral theory; p-adic q-analysis; fractional calculus and its applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Following the success of the second Special Issue of Symmetry, entitled “Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II”, it is my pleasure to be the Guest Editor for Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition.

Special functions and orthogonal polynomials, in particular, have been around for centuries. In the twentieth century, the emphasis was on special functions satisfying linear differential equations, but this has been extended to difference equations, partial differential equations, and nonlinear differential equations. The theory of the symmetries of special functions, orthogonal polynomials, and differential equations is well improved, their relations to integrability are known, and there are many corresponding results and applications. They provide us with the means to compute the symmetries of a given equation in an algorithmic manner and, most importantly, to implement it in symbolic computations.

This Special Issue will reflect the diversity of topics across the world. The Special Issue’s contributions will cover the symmetries of difference equations, discrete dynamical systems, special functions, orthogonal polynomials, symmetries, and integrable difference equations.

Dr. Serkan Araci
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

  • orthogonal polynomials
  • difference equations
  • symmetries in special functions
  • symmetries in orthogonal polynomials
  • symmetries of difference equations
  • the analytical properties and applications of special functions
  • inequalities for special functions
  • the integration of the products of special functions
  • the properties of ordinary and general families of special polynomials
  • operational techniques involving special polynomials
  • classes of mixed special polynomials and their properties
  • other miscellaneous applications of special functions and special polynomials

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Published Papers (7 papers)

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Research

26 pages, 309 KiB  
Article
Overview of Six Number/Polynomial Sequences Defined by Quadratic Recurrence Relations
by Yujie Kang, Marta Na Chen and Wenchang Chu
Symmetry 2025, 17(5), 714; https://doi.org/10.3390/sym17050714 - 7 May 2025
Viewed by 110
Abstract
Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and [...] Read more.
Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
18 pages, 319 KiB  
Article
Some New Inequalities for the Gamma and Polygamma Functions
by Waad Al Sayed and Hesham Moustafa
Symmetry 2025, 17(4), 595; https://doi.org/10.3390/sym17040595 - 14 Apr 2025
Viewed by 207
Abstract
In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate their complete monotonicity. Also, we investigate their completely monotonic degrees. This concept gives [...] Read more.
In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate their complete monotonicity. Also, we investigate their completely monotonic degrees. This concept gives more accuracy in measuring the complete monotonicity property. These new bounds are better than some of the recently published results. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
13 pages, 274 KiB  
Article
Some New Bounds for Bateman’s G-Function in Terms of the Digamma Function
by Hesham Moustafa and Waad Al Sayed
Symmetry 2025, 17(4), 563; https://doi.org/10.3390/sym17040563 - 8 Apr 2025
Viewed by 191
Abstract
In this paper, we present some new symmetric bounds for Bateman’s G-function and its derivatives, in terms of the digamma and polygamma functions, which are better than some recent results. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
19 pages, 339 KiB  
Article
A New Generalization of q-Laguerre-Based Appell Polynomials and Quasi-Monomiality
by Naeem Ahmad and Waseem Ahmad Khan
Symmetry 2025, 17(3), 439; https://doi.org/10.3390/sym17030439 - 14 Mar 2025
Viewed by 394
Abstract
In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q [...] Read more.
In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q-Bessel–Tricomi functions. These polynomials are studied by means of generating function, series expansion, and determinant representation. Also, these polynomials are further examined within the framework of q-quasi-monomiality, leading to the establishment of essential operational identities. We then derive operational representations, as well as q-differential equations for the three-variable q-Laguerre-based Appell polynomials. Some examples are constructed in terms of q-Laguerre–Hermite-based Bernoulli, Euler, and Genocchi polynomials in order to illustrate the main results. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
24 pages, 344 KiB  
Article
On a General Functional Equation
by Anna Bahyrycz
Symmetry 2025, 17(3), 320; https://doi.org/10.3390/sym17030320 - 20 Feb 2025
Viewed by 421
Abstract
In this paper, we deal with a general functional equation in several variables. We prove the hyperstability of this equation in (m + 1)-normed spaces and describe its general solution in some special cases. In this way, we solve the problems posed [...] Read more.
In this paper, we deal with a general functional equation in several variables. We prove the hyperstability of this equation in (m + 1)-normed spaces and describe its general solution in some special cases. In this way, we solve the problems posed by Ciepliński. The considered equation was introduced as a generalization of the equation characterizing n-quadratic functions and has symmetric coefficients (up to sign), and it also generalizes many other known functional equations with symmetric coefficients, such as the multi-Cauchy equation, the multi-Jensen equation, and the multi-Cauchy–Jensen equation. Our results generalize several known results. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
16 pages, 294 KiB  
Article
Characterization of Bi-Starlike Functions: A Daehee Polynomial Approach
by Timilehin Gideon Shaba, Serkan Araci, Babatunde Olufemi Adebesin, Fuat Usta and Bilal Khan
Symmetry 2024, 16(12), 1640; https://doi.org/10.3390/sym16121640 - 11 Dec 2024
Viewed by 762
Abstract
This research investigates the second Hankel determinant for a specific class of functions associated with the Daehee polynomial. To achieve this, we introduce new subclasses of starlike functions in the context of Daehee polynomials. In complex analysis, establishing precise bounds for coefficient estimates [...] Read more.
This research investigates the second Hankel determinant for a specific class of functions associated with the Daehee polynomial. To achieve this, we introduce new subclasses of starlike functions in the context of Daehee polynomials. In complex analysis, establishing precise bounds for coefficient estimates in bi-univalent functions is essential, as these coefficients define the fundamental properties of conformal mappings. In this study, we derive sharp bounds for coefficient estimates within new subclasses of starlike functions related to Daehee polynomials, with most of the obtained limits demonstrating high accuracy. This work aims to inspire further exploration of rigorous bounds for analytic functions associated with innovative mapping domains. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
14 pages, 1437 KiB  
Article
Hyperbolic Non-Polynomial Spline Approach for Time-Fractional Coupled KdV Equations: A Computational Investigation
by Miguel Vivas-Cortez, Majeed A. Yousif, Pshtiwan Othman Mohammed, Alina Alb Lupas, Ibrahim S. Ibrahim and Nejmeddine Chorfi
Symmetry 2024, 16(12), 1610; https://doi.org/10.3390/sym16121610 - 4 Dec 2024
Cited by 6 | Viewed by 799
Abstract
The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable [...] Read more.
The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)
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