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Axioms
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Special Functions and Polynomials: Theory, Practice, Applications, and Modeling
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Special Functions and Polynomials: Theory, Practice, Applications, and Modeling

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 8984

Special Issue Editors

Dr. Serkan Araci

E-Mail Website
Guest Editor
Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Turkey
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
Prof. Dr. Mehmet Acikgoz

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Arts and Sciences, Gaziantep University, TR-27310 Gaziantep, Turkey
Interests: q-analysis; functional analysis; p-adic analysis; analytic numbers theory; special functions
Dr. Ugur Duran

E-Mail Website
Guest Editor
Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey
Interests: special functions; q-calculus; p-adic analysis; umbral calculus; special polynomials

Special Issue Information

Dear Colleagues,

Special polynomials and numbers have significant roles in various branches of mathematics, theoretical physics, and engineering. The problems arising in mathematical physics and engineering are framed in terms of differential equations. Most of these equations can only be treated using various families of special polynomials which provide new means of mathematical analysis. They are widely used in computational models of scientific and engineering problems. Applications of various properties of special functions and polynomials also arise in problems of number theory, combinatorics, theoretical physics, and other areas of pure and applied mathematics, providing the motivation for introducing a new class of special functions and polynomials.

Each paper that will be published in this Special Issue should aim to enrich our understanding of current research problems, theories, applications, and modeling of special functions and polynomials.

Potential topics include but are not limited to the following:

  • Special functions
  • Special polynomials
  • Orthogonal polynomials
  • q-Analiz
  • p-adic analysis
  • Umbral calculus
  • Differential and integral equations
  • Number theory

Dr. Serkan Araci
Prof. Dr. Mehmet Acikgoz
Dr. Ugur Duran
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. Axioms 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 (6 papers)

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Research

16 pages, 333 KiB  
Article
On the Use of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions for the Calculation of Infinite Sums and the Analysis of Zeroes of Analytical Functions
by Sergey Sekatskii
Axioms 2023, 12(1), 68; https://doi.org/10.3390/axioms12010068 - 07 Jan 2023
Cited by 2 | Viewed by 1053
Abstract
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and [...] Read more.
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. On many occasions, this enables to facilitate the obtaining of known results thus having important methodological meaning. Additionally, some new results, to the best of our knowledge, are also obtained in this way. For example, we established new properties of the sum of inverse zeroes of a digamma function, new formulae for the sums ∑kiρi2 for zeroes ρi of incomplete gamma and Riemann zeta functions having the order ki (These results can be straightforwardly generalized for the sums ∑kiρin with integer n > 2, and so on.) Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
22 pages, 415 KiB  
Article
Application of Orthogonal Polynomial in Orthogonal Projection of Algebraic Surface
by Xudong Wang, Xiaowu Li and Yuxia Lyu
Axioms 2022, 11(10), 544; https://doi.org/10.3390/axioms11100544 - 11 Oct 2022
Viewed by 1170
Abstract
Point orthogonal projection onto an algebraic surface is a very important topic in computer-aided geometric design and other fields. However, implementing this method is currently extremely challenging and difficult because it is difficult to achieve to desired degree of robustness. Therefore, we construct [...] Read more.
Point orthogonal projection onto an algebraic surface is a very important topic in computer-aided geometric design and other fields. However, implementing this method is currently extremely challenging and difficult because it is difficult to achieve to desired degree of robustness. Therefore, we construct an orthogonal polynomial, which is the ninth formula, after the inner product of the eighth formula itself. Additionally, we use the Newton iterative method for the iteration. In order to ensure maximum convergence, two techniques are used before the Newton iteration: (1) Newton’s gradient descent method, which is used to make the initial iteration point fall on the algebraic surface, and (2) computation of the foot-point and moving the iterative point to the close position of the orthogonal projection point of the algebraic surface. Theoretical analysis and experimental results show that the proposed algorithm can accurately, efficiently, and robustly converge to the orthogonal projection point for test points in different spatial positions. Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
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14 pages, 319 KiB  
Article
Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains
by Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, Isra Al-shbeil, Sarfraz Ahmad, Ajmal Khan, Bilal Khan and Jianhua Gong
Axioms 2022, 11(10), 494; https://doi.org/10.3390/axioms11100494 - 23 Sep 2022
Cited by 9 | Viewed by 934
Abstract
In the current work, by using the familiar q-calculus, first, we study certain generalized conic-type regions. We then introduce and study a subclass of the multivalent q-starlike functions that map the open unit disk into the generalized conic domain. Next, we [...] Read more.
In the current work, by using the familiar q-calculus, first, we study certain generalized conic-type regions. We then introduce and study a subclass of the multivalent q-starlike functions that map the open unit disk into the generalized conic domain. Next, we study potentially effective outcomes such as sufficient restrictions and the Fekete–Szegö type inequalities. We attain lower bounds for the ratio of a good few functions related to this lately established class and sequences of the partial sums. Furthermore, we acquire a number of attributes of the corresponding class of q-starlike functions having negative Taylor–Maclaurin coefficients, including distortion theorems. Moreover, various important corollaries are carried out. The new explorations appear to be in line with a good few prior commissions and the current area of our recent investigation. Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
11 pages, 284 KiB  
Article
Nonoscillation and Oscillation Criteria for a Class of Second-Order Nonlinear Neutral Delay Differential Equations with Positive and Negative Coefficients
by Rongrong Guo, Qingdao Huang and Haifeng Tian
Axioms 2022, 11(6), 281; https://doi.org/10.3390/axioms11060281 - 10 Jun 2022
Viewed by 1266
Abstract
In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent [...] Read more.
In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent results provided in the literature. Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
16 pages, 302 KiB  
Article
General Opial Type Inequality and New Green Functions
by Ana Gudelj, Kristina Krulić Himmelreich and Josip Pečarić
Axioms 2022, 11(6), 252; https://doi.org/10.3390/axioms11060252 - 26 May 2022
Viewed by 1319
Abstract
In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space (Ω,Σ,μ). We give an [...] Read more.
In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space (Ω,Σ,μ). We give an new result involving four new Green functions. Our results include Grüss and Ostrowski type inequalities related to the generalized Opial type inequality. The obtained inequalities are of Opial type because the integrals contain the function and its integral representation. They are not a direct generalization of the Opial inequality. Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
10 pages, 284 KiB  
Article
Generalized k-Fractional Chebyshev-Type Inequalities via Mittag-Leffler Functions
by Zhiqiang Zhang, Ghulam Farid, Sajid Mehmood, Chahn-Yong Jung and Tao Yan
Axioms 2022, 11(2), 82; https://doi.org/10.3390/axioms11020082 - 21 Feb 2022
Cited by 4 | Viewed by 1407
Abstract
Mathematical inequalities have gained importance and popularity due to the application of integral operators of different types. The present paper aims to give Chebyshev-type inequalities for generalized k-integral operators involving the Mittag-Leffler function in kernels. Several new results can be deduced for [...] Read more.
Mathematical inequalities have gained importance and popularity due to the application of integral operators of different types. The present paper aims to give Chebyshev-type inequalities for generalized k-integral operators involving the Mittag-Leffler function in kernels. Several new results can be deduced for different integral operators, along with Riemann–Liouville fractional integrals by substituting convenient parameters. Moreover, the presented results generalize several already published inequalities. Full article
(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)
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