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Axioms
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Orthogonal Polynomials, Special Functions and Applications
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Orthogonal Polynomials, Special Functions and Applications

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

Deadline for manuscript submissions: closed (1 December 2022) | Viewed by 32358

Special Issue Editor

Prof. Dr. Gradimir V. Milovanović

E-Mail Website
Guest Editor
1. Serbian Academy of Sciences and Arts, Kneza Mihaila 35, 11000 Belgrade, Serbia
2. Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
Interests: orthogonal polynomials, orthogonal systems and special functions; interpolation, quadrature processes and integral equations; approximations by polynomials, splines and linear operators; numerical and optimization methods; polynomials (extremal problems, inequalities, zeros); iterative processes and inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Orthogonal polynomials and orthogonal functions, as well as other special functions, are gaining in importance everyday and their development is often conditioned by their application in many areas of applied and computational sciences. This Special Issue of Axioms is devoted to various aspects of the theory of orthogonality in real or complex spaces with respect to the standard inner product (classical and strongly nonclassical cases) and moment functionals, including one-dimensional and multidimensional cases. Contributions considering the development and application of special functions, as well as problems in which special functions play a significant role, are welcome. Particularly interesting are theories and applications in which both orthogonality and special functions are represented. Consideration of problems in which special functions play a significant role, as well as applications of orhogonal polynomials in approximation theory in the broadest sense, including quadrature formulas and integral equations, will be particularly appreciated. Furthermore, applications and algorithms for solving open problems in mathematics, physics, and technical sciences are of interest. Our goal is to gather experts, as well as young researchers focused on the same task, in order to promote and exchange knowledge and improve communication and application. We invite research papers, as well as review articles.

Prof. Dr. Gradimir V. Milovanović
Guest Editor

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Keywords

  • Orthogonal polynomials
  • Special functions
  • Zeros
  • Recurrence relations
  • Inner product
  • Quadrature formulas
  • Integral equations
  • Approximation of functions
  • Generating functions
  • Asymptotics
  • Inequalities

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Related Special Issue

Published Papers (17 papers)

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Research

4 pages, 226 KiB  
Article
A Note on Generalization of Combinatorial Identities Due to Gould and Touchard
by Arjun K. Rathie and Dongkyu Lim
Axioms 2023, 12(3), 268; https://doi.org/10.3390/axioms12030268 - 5 Mar 2023
Cited by 2 | Viewed by 1218
Abstract
Using a hypergeometric series approach, a general combinatorial identity is found in this note, and among its special cases are well-known and classical combinatorial identities due to Gould and Touchard. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
12 pages, 323 KiB  
Article
Pole Allocation for Rational Gauss Quadrature Rules for Matrix Functionals Defined by a Stieltjes Function
by Jihan Alahmadi, Miroslav Pranić and Lothar Reichel
Axioms 2023, 12(2), 105; https://doi.org/10.3390/axioms12020105 - 19 Jan 2023
Viewed by 1311
Abstract
This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a [...] Read more.
This paper considers the computation of approximations of matrix functionals of form F(A):=vTf(A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The functional F(A) is approximated by a rational Gauss quadrature rule with poles on the negative real axis (or part thereof) in the complex plane, and we focus on the allocation of the poles. Specifically, we propose that the poles, when considered positive point charges, be allocated to make the negative real axis (or part thereof) approximate an equipotential curve. This is easily achieved with the aid of conformal mapping. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
27 pages, 511 KiB  
Article
On Perfectness of Systems of Weights Satisfying Pearson’s Equation with Nonstandard Parameters
by Alexander Aptekarev, Alexander Dyachenko and Vladimir Lysov
Axioms 2023, 12(1), 89; https://doi.org/10.3390/axioms12010089 - 15 Jan 2023
Cited by 2 | Viewed by 1326
Abstract
Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves [...] Read more.
Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves in C. Some applications lead to multiple orthogonality with respect to a number of such measures. For a system of r orthogonality measures, the perfectness is an important property: in particular, it implies the uniqueness for the whole family of corresponding multiple orthogonal polynomials and the (r+2)-term recurrence relations. In this paper, we introduce a unified approach which allows to prove the perfectness of the systems of complex measures satisfying Pearson’s equation with nonstandard parameters. We also study the polynomials satisfying multiple orthogonality relations with respect to a system of discrete measures. The well-studied families of multiple Charlier, Krawtchouk, Meixner and Hahn polynomials correspond to the systems of measures defined by the difference Pearson’s equation with standard real parameters. Using the same approach, we verify the perfectness of such systems for general parameters. For some values of the parameters, discrete measures should be replaced with the continuous measures with non-real supports. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
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46 pages, 539 KiB  
Article
Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials
by Peter Junghanns and Robert Kaiser
Axioms 2023, 12(1), 55; https://doi.org/10.3390/axioms12010055 - 3 Jan 2023
Cited by 2 | Viewed by 1135
Abstract
In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is [...] Read more.
In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
18 pages, 430 KiB  
Article
Conditional Expanding of Functions by q-Lidstone Series
by Maryam Al-Towailb and Zeinab S. I. Mansour
Axioms 2023, 12(1), 22; https://doi.org/10.3390/axioms12010022 - 25 Dec 2022
Cited by 2 | Viewed by 1328
Abstract
This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
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11 pages, 320 KiB  
Article
Hilbert’s Double Series Theorem’s Extensions via the Mathieu Series Approach
by Tibor K. Pogány
Axioms 2022, 11(11), 643; https://doi.org/10.3390/axioms11110643 - 14 Nov 2022
Viewed by 1532
Abstract
The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result is extended in the case of a not exhaustively investigated non-homogeneous kernel, [...] Read more.
The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result is extended in the case of a not exhaustively investigated non-homogeneous kernel, which mutually covers the homogeneous kernel cases as well. Particularly, novel Hilbert’s double series inequality results are presented, which include the upper bounds built exclusively with non-weighted ℓp–norms. The main mathematical tools are the integral expression of Mathieu (a,λ)-series, the Hölder inequality and a generalization of the double series theorem by Yang. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
17 pages, 1369 KiB  
Article
Estimation of Truncation Error in Statistical Description of Communication Signals over mm-Wave Channels
by Zvezdan Marjanović, Dejan N. Milić and Goran T. Đorđević
Axioms 2022, 11(10), 569; https://doi.org/10.3390/axioms11100569 - 19 Oct 2022
Cited by 2 | Viewed by 1773
Abstract
This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density [...] Read more.
This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density function (PDF) and cumulative distribution function (CDF) of the composite signal envelope over an mm-Wave channel. The formulas for the PDF and CDF are expressed in the convergent infinity series form. The main contribution of the paper is in estimating the upper bounds for absolute truncation error in evaluating PDF and CDF of the signal envelope. We also derive the formulas for the required number of terms in the summation under the condition of achieving a given accuracy for typical values of channel parameters. In deriving these formulas, we use the alternating series estimation theorem, as well as some properties of orthogonal polynomials in order to derive upper bounds for hypergeometric functions. Based on the newly derived formulas, numerical results are presented and commented upon. The analytical results are verified by Monte Carlo simulations. The results are essential in the designing and performance estimating of the fifth-generation (5G) and beyond wireless networks. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
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15 pages, 327 KiB  
Article
Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials
by Esra Güldoğan Lekesiz, Rabia Aktaş and Iván Area
Axioms 2022, 11(10), 558; https://doi.org/10.3390/axioms11100558 - 14 Oct 2022
Cited by 2 | Viewed by 1755
Abstract
Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s [...] Read more.
Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
11 pages, 269 KiB  
Article
Resolution of an Isolated Case of a Quadratic Hypergeometric 2F1 Transformation
by Mohamed Jalel Atia
Axioms 2022, 11(10), 533; https://doi.org/10.3390/axioms11100533 - 5 Oct 2022
Cited by 2 | Viewed by 1474
Abstract
The identity [...] Read more.
The identity 2F1(α,β;2α;z)=(1−z2)−β2F1(β2,β+12;α+12;(z2−z)2) given, either by I.S. Gradshteyn and I.M. Ryzhik in Table of integrals series and products named 9.134 or in the handbook “mathematical functions with formulas, graphs and mathematical tables” done by Abramowitz-Stegun named 15.3.20 or in the book “special functions” done by G. Andrews, R. Askey and R. Roy named 3.1.7 page 127 with a slight modification is true provided that {2α+1, α+32} are not natural numbers and α−β is not an integer (see Gradshteyn, Ryzhik, 9.130). In this manuscript we consider a case where α−β is an integer by taking β=2a, α=−n+1. We give and prove the right identity for any positive integer a and for any any positive integer n. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
19 pages, 309 KiB  
Article
Piecewise Fractional Jacobi Polynomial Approximations for Volterra Integro-Differential Equations with Weakly Singular Kernels
by Haiyang Li and Junjie Ma
Axioms 2022, 11(10), 530; https://doi.org/10.3390/axioms11100530 - 4 Oct 2022
Cited by 2 | Viewed by 1585
Abstract
This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the [...] Read more.
This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the existence, uniqueness and convergence properties of Galerkin solutions by exploiting the decaying rate of the coefficients of the transformed fractional Jacobi series. Finally, numerical experiments are carried out to illustrate the performance of the piecewise Galerkin solution. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
10 pages, 276 KiB  
Article
Degenerate Fubini-Type Polynomials and Numbers, Degenerate Apostol–Bernoulli Polynomials and Numbers, and Degenerate Apostol–Euler Polynomials and Numbers
by Siqintuya Jin, Muhammet Cihat Dağli and Feng Qi
Axioms 2022, 11(9), 477; https://doi.org/10.3390/axioms11090477 - 17 Sep 2022
Cited by 2 | Viewed by 2191
Abstract
In this paper, by introducing degenerate Fubini-type polynomials, with the help of the Faà di Bruno formula and some properties of partial Bell polynomials, the authors provide several new explicit formulas and recurrence relations for Fubini-type polynomials and numbers, associate the newly defined [...] Read more.
In this paper, by introducing degenerate Fubini-type polynomials, with the help of the Faà di Bruno formula and some properties of partial Bell polynomials, the authors provide several new explicit formulas and recurrence relations for Fubini-type polynomials and numbers, associate the newly defined degenerate Fubini-type polynomials with degenerate Apostol–Bernoulli polynomials and degenerate Apostol–Euler polynomials of order α. These results enable one to present additional relations for some degenerate special polynomials and numbers. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
14 pages, 292 KiB  
Article
On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials
by Mohamed Jalel Atia and Majed Benabdallah
Axioms 2022, 11(7), 344; https://doi.org/10.3390/axioms11070344 - 19 Jul 2022
Cited by 1 | Viewed by 1650
Abstract
In this paper, we first give some results on monic generalized Hermite polynomials (GHP) {Hn(μ)(x)}n≥0, orthogonal with respect to the positive weight [...] Read more.
In this paper, we first give some results on monic generalized Hermite polynomials (GHP) {Hn(μ)(x)}n≥0, orthogonal with respect to the positive weight |x|2μe−x2,μ>−12,x∈R, which will lead to the formulation of the second-order spectralvectorial differential equation (SVDE) that the GHP satisfies. This SVDE differs from the one given in G. Szego (problem 25. p. 380), which is a pseudo-spectral equation. Second, we give the SVDE, as conjecture, satisfied by the generalized Jacobi polynomials Jn(α,α+1)(x,μ), orthogonal with respect to the positive weight w(x,α;μ)=|x|−μ(1−x2)α(1−x), μ<1, α>−1 on [−1,1]. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
19 pages, 363 KiB  
Article
A Family of Generalized Legendre-Based Apostol-Type Polynomials
by Talha Usman, Nabiullah Khan, Mohd Aman and Junesang Choi
Axioms 2022, 11(1), 29; https://doi.org/10.3390/axioms11010029 - 14 Jan 2022
Cited by 6 | Viewed by 2413
Abstract
Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate [...] Read more.
Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
5 pages, 244 KiB  
Article
On the Truncated Multidimensional Moment Problems in Cn
by Sergey Zagorodnyuk
Axioms 2022, 11(1), 20; https://doi.org/10.3390/axioms11010020 - 5 Jan 2022
Cited by 1 | Viewed by 1336
Abstract
We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite [...] Read more.
We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
16 pages, 366 KiB  
Article
Generalized Summation Formulas for the Kampé de Fériet Function
by Junesang Choi, Gradimir V. Milovanović and Arjun K. Rathie
Axioms 2021, 10(4), 318; https://doi.org/10.3390/axioms10040318 - 25 Nov 2021
Cited by 9 | Viewed by 2586
Abstract
By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with [...] Read more.
By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2F1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the 2F1 due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
6 pages, 243 KiB  
Article
On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r
by Pavel Trojovský
Axioms 2021, 10(3), 144; https://doi.org/10.3390/axioms10030144 - 1 Jul 2021
Cited by 4 | Viewed by 2893
Abstract
For r≥2 and a≥1 integers, let (tn(r,a))n≥1 be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence [...] Read more.
For r≥2 and a≥1 integers, let (tn(r,a))n≥1 be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence tn(r,a)=tn−1(r,a)+⋯+tn−r(r,a) for n>r, with initial values ti(r,a)=1, for all i∈[1,r−1] and tr(r,a)=a. In this paper, we shall prove (in particular) that, for any given r≥2, there exists a positive proportion of positive integers which can not be written as tn(r,a) for any (n,a)∈Z≥r+2×Z≥1. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
23 pages, 345 KiB  
Article
Spectral Transformations and Associated Linear Functionals of the First Kind
by Juan Carlos García-Ardila and Francisco Marcellán
Axioms 2021, 10(2), 107; https://doi.org/10.3390/axioms10020107 - 28 May 2021
Cited by 1 | Viewed by 2509
Abstract
Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation [...] Read more.
Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
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