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Orthogonal Polynomials and Special Functions
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Orthogonal Polynomials and Special Functions

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 14006

Special Issue Editor

Prof. Dr. Clemente Cesarano

E-Mail Website
Guest Editor
Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; Lie algebra
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The theory of generalized orthogonal polynomials and special functions is applied in different branches of pure and applied mathematics, as well as in physics. A combination of techniques involving methods of an algebraic nature and numerical methods may offer a powerful tool to solve problems in pure and applied mathematics. In the last years, the combined use of operational methods, orthogonal polynomials, and special functions has provided solutions that are hardly achievable with conventional means. Furthermore, the structural properties of polynomials in the framework of standard L2 orthogonality with respect to a Borel measure (or a weight function) have been deeply studied for other patterns of orthogonality, like multiple orthogonal polynomials, orthogonal polynomials in several variables, or Sobolev orthogonal polynomials.

Prof. Dr. Clemente Cesarano
Guest Editor

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Published Papers (8 papers)

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Research

18 pages, 326 KiB  
Article
Solvability of a State–Dependence Functional Integro-Differential Inclusion with Delay Nonlocal Condition
by Taher S. Hassan, Reda Gamal Ahmed, Ahmed M. A. El-Sayed, Rami Ahmad El-Nabulsi, Osama Moaaz and Mouataz Billah Mesmouli
Mathematics 2022, 10(14), 2420; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142420 - 11 Jul 2022
Cited by 3 | Viewed by 1075
Abstract
There is great focus on phenomena that depend on their past history or their past state. The mathematical models of these phenomena can be described by differential equations of a self-referred type. This paper is devoted to studying the solvability of a state-dependent [...] Read more.
There is great focus on phenomena that depend on their past history or their past state. The mathematical models of these phenomena can be described by differential equations of a self-referred type. This paper is devoted to studying the solvability of a state-dependent integro-differential inclusion. The existence and uniqueness of solutions to a state-dependent functional integro-differential inclusion with delay nonlocal condition is studied. We, moreover, conclude the existence of solutions to the problem with the integral condition and the infinite-point boundary one. Some properties of the solutions are given. Finally, two examples illustrating the main result are presented. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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21 pages, 351 KiB  
Article
Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function
by Shilpi Jain, Rahul Goyal, Praveen Agarwal, Antonella Lupica and Clemente Cesarano
Mathematics 2021, 9(22), 2944; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222944 - 18 Nov 2021
Cited by 5 | Viewed by 1474
Abstract
The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these [...] Read more.
The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
18 pages, 309 KiB  
Article
Some New Results on Bicomplex Bernstein Polynomials
by Carlo Cattani, Çíğdem Atakut, Özge Özalp Güller and Seda Karateke
Mathematics 2021, 9(21), 2748; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212748 - 29 Oct 2021
Viewed by 1311
Abstract
The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate [...] Read more.
The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
18 pages, 564 KiB  
Article
Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind
by Domenico Mezzanotte, Donatella Occorsio and Maria Grazia Russo
Mathematics 2021, 9(21), 2652; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212652 - 20 Oct 2021
Cited by 4 | Viewed by 1179
Abstract
In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The [...] Read more.
In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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10 pages, 320 KiB  
Article
Discrete Hypergeometric Legendre Polynomials
by Tom Cuchta and Rebecca Luketic
Mathematics 2021, 9(20), 2546; https://0-doi-org.brum.beds.ac.uk/10.3390/math9202546 - 11 Oct 2021
Cited by 5 | Viewed by 1612
Abstract
A discrete analog of the Legendre polynomials defined by discrete hypergeometric series is investigated. The resulting polynomials have qualitatively similar properties to classical Legendre polynomials. We derive their difference equations, recurrence relations, and generating function. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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15 pages, 283 KiB  
Article
A General Family of q-Hypergeometric Polynomials and Associated Generating Functions
by Hari Mohan Srivastava and Sama Arjika
Mathematics 2021, 9(11), 1161; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111161 - 21 May 2021
Cited by 16 | Viewed by 1619
Abstract
Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of [...] Read more.
Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
10 pages, 309 KiB  
Article
On the Oscillatory Properties of Solutions of Second-Order Damped Delay Differential Equations
by Awatif A. Hendi, Osama Moaaz, Clemente Cesarano, Wedad R. Alharbi and Mohamed A. Abdou
Mathematics 2021, 9(9), 1060; https://0-doi-org.brum.beds.ac.uk/10.3390/math9091060 - 09 May 2021
Cited by 4 | Viewed by 1728
Abstract
In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example [...] Read more.
In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example is presented in order to illustrate the significance of the results. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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21 pages, 306 KiB  
Article
New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials
by Waleed Mohamed Abd-Elhameed and Afnan Ali
Mathematics 2021, 9(1), 74; https://0-doi-org.brum.beds.ac.uk/10.3390/math9010074 - 31 Dec 2020
Cited by 7 | Viewed by 1474
Abstract
The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms [...] Read more.
The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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