Orthogonal Polynomials, Special Functions and Applications: 2nd Edition

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

Deadline for manuscript submissions: closed (31 December 2024) | Viewed by 8621

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Guest Editor
Mathematical Institute, Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia
Interests: approximation theory; special functions; extremal problems; inequalities; orthogonal polynomials; nonclassical orthogonal polynomials; numerical analysis; numerical linear algebra; interpolation in complex plane; orthogonality on the semicircle
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Special Issue Information

Dear Colleagues,

This Special Issue is a continuation of the previous successful Special Issue “Orthogonal Polynomials, Special Functions and Applications”.

Orthogonal polynomials and orthogonal functions, as well as other special functions, are gaining increasing importance 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 products (classical and strongly non-classical cases) and moment functionals, including one-dimensional and multidimensional cases, as well as to orthogonalization in numerical linear algebra. Contributions that consider the development and application of special functions, as well as problems in which special functions play a significant role, are welcome. Particularly interesting are the theories and applications in which both orthogonality and special functions are represented. Consideration of the problems in which special functions play a significant role, as well as applications of orthogonal polynomials in approximation theory in the broadest sense, including quadrature formulas and integral equations, will be particularly appreciated. Furthermore,  spectral, collocation and related methods for initial value and initial-boundary value problems that involve PDEs, as well as 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 the submission of research papers, as well as review articles.

Prof. Dr. Gradimir V. Milovanović
Guest Editor

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Keywords

  • orthogonal polynomials and functions
  • orthogonalization in numerical linear algebra
  • special functions
  • hypergeometric functions
  • mittag-leffler functions and generalizations
  • zeros
  • recurrence relations
  • inner products
  • numerical integration
  • quadrature and cubature formulas
  • numerical summation of series
  • numerical differentiation
  • integral equations
  • numerical methods for integral equations and transforms
  • approximation of functions
  • spline approximation
  • padé approximation
  • weighted approximation
  • spectral, collocation and related methods for bvp problems
  • generating functions
  • asymptotics
  • inequalities

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Published Papers (10 papers)

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Research

43 pages, 521 KiB  
Article
On Finite Exceptional Orthogonal Polynomial Sequences Composed of Rational Darboux Transforms of Romanovski-Jacobi Polynomials
by Gregory Natanson
Axioms 2025, 14(3), 218; https://doi.org/10.3390/axioms14030218 - 16 Mar 2025
Viewed by 205
Abstract
The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. [...] Read more.
The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a) exactly quantized by the EOPs in question. Full article
17 pages, 1187 KiB  
Article
Müntz–Legendre Wavelet Collocation Method for Solving Fractional Riccati Equation
by Fatemeh Soleyman and Iván Area
Axioms 2025, 14(3), 185; https://doi.org/10.3390/axioms14030185 - 2 Mar 2025
Viewed by 388
Abstract
We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By [...] Read more.
We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article
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19 pages, 982 KiB  
Article
Error Estimators for a Krylov Subspace Iterative Method for Solving Linear Systems of Equations with a Symmetric Indefinite Matrix
by Mohammed Alibrahim, Mohammad Taghi Darvishi, Lothar Reichel and Miodrag M. Spalević
Axioms 2025, 14(3), 179; https://doi.org/10.3390/axioms14030179 - 28 Feb 2025
Viewed by 325
Abstract
This paper describes a Krylov subspace iterative method designed for solving linear systems of equations with a large, symmetric, nonsingular, and indefinite matrix. This method is tailored to enable the evaluation of error estimates for the computed iterates. The availability of error estimates [...] Read more.
This paper describes a Krylov subspace iterative method designed for solving linear systems of equations with a large, symmetric, nonsingular, and indefinite matrix. This method is tailored to enable the evaluation of error estimates for the computed iterates. The availability of error estimates makes it possible to terminate the iterative process when the estimated error is smaller than a user-specified tolerance. The error estimates are calculated by leveraging the relationship between the iterates and Gauss-type quadrature rules. Computed examples illustrate the performance of the iterative method and the error estimates. Full article
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10 pages, 273 KiB  
Article
Krein–Sobolev Orthogonal Polynomials II
by Alexander Jones, Lance Littlejohn and Alejandro Quintero Roba
Axioms 2025, 14(2), 115; https://doi.org/10.3390/axioms14020115 - 1 Feb 2025
Viewed by 474
Abstract
In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials {Kn}n=0—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space H1[1,1] with respect to [...] Read more.
In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials {Kn}n=0—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space H1[1,1] with respect to the (positive-definite) inner product (f,g)1,c:=f(1)f(1)g¯(1)g¯(1)2+11(f(x)g¯(x)+cf(x)g¯(x))dx, where c is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator Kc(c>0) in L2(1,1). Other than K0 and K1, these polynomials are not eigenfunctions of Kc. As shown by Littlejohn and Quintero, the sequence {Kn}n=0 forms a complete orthogonal set in the first left-definite space (H1[1,1],(·,·)1,c) associated with (Kc,L2(1,1)). Furthermore, they show that, for n1,Kn(x) has n distinct zeros in (1,1). In this note, we find an explicit formula for Krein–Sobolev polynomials {Kn}n=0. Full article
33 pages, 737 KiB  
Article
Orthogonal Polynomials on Radial Rays in the Complex Plane: Construction, Properties and Applications
by Gradimir V. Milovanović
Axioms 2025, 14(1), 65; https://doi.org/10.3390/axioms14010065 - 16 Jan 2025
Viewed by 623
Abstract
Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results [...] Read more.
Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results and examples. In addition to several types of standard orthogonality, the concept of orthogonality on arbitrary radial rays is introduced, some or all of which may be infinite. A general method for numerical constructing, the so-called discretized Stieltjes–Gautschi procedure, is described and several interesting examples are presented. The main properties, zero distribution and some applications are also given. Special attention is paid to completely symmetric cases. Recurrence relations for such kinds of orthogonal polynomials and their zero distribution, as well as a connection with the standard polynomials orthogonal on the real line, are derived, including the corresponding linear differential equation of the second order. Finally, some applications in physics and electrostatics are mentioned. Full article
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12 pages, 270 KiB  
Article
Extension of Chu–Vandermonde Identity and Quadratic Transformation Conditions
by Mohamed Jalel Atia and Maged Alkilayh
Axioms 2024, 13(12), 825; https://doi.org/10.3390/axioms13120825 - 25 Nov 2024
Cited by 1 | Viewed by 736
Abstract
In 1812, Gauss stated the following identity: [...] Read more.
In 1812, Gauss stated the following identity: F12(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)Γ(cb), where, in the real case, cab>0 and as an immediat consequence the Chu–Vandermonde identity: F12(a,n;c;1)=(ca)n(c)n for any positive integer n. In this paper, we investigate the case when cab<0 by taking c=2b=2n, n and a are positive integers (cab=na<0). We give two significant applications stemming from these findings. The second part of the paper will be devoted to Kummer’s conditions concerning hypergeometric quadratic transformations, particularly focusing on the distinctions between the conditions provided by Gradshteyn and Ryzhik (GR) and those by Erdélyi, Magnus, Oberhettinger, and Tricomi (EMOI) are outlined. We establish that the conditions given by GR differ from those of EMOI, and we explore the methodologies employed by both groups in deriving their results. This leads us to conclude that the search for exact and unified conditions remains an open problem. Full article
20 pages, 465 KiB  
Article
A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation
by Maria Carmela De Bonis and Donatella Occorsio
Axioms 2024, 13(11), 750; https://doi.org/10.3390/axioms13110750 - 30 Oct 2024
Viewed by 1323
Abstract
In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α [...] Read more.
In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(mα)0y(yx)mα1f(m)(x)dx,y>0, with m1<αm,mN. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. Full article
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17 pages, 354 KiB  
Article
On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities
by Rakesh K. Parmar, Tibor K. Pogány and Uthara Sabu
Axioms 2024, 13(8), 534; https://doi.org/10.3390/axioms13080534 - 7 Aug 2024
Viewed by 856
Abstract
The principal aim of this paper is to introduce the extended Voigt-type function Vμ,ν(x,y) and its counterpart extension Wμ,ν(x,y), involving the Neumann function Yν in [...] Read more.
The principal aim of this paper is to introduce the extended Voigt-type function Vμ,ν(x,y) and its counterpart extension Wμ,ν(x,y), involving the Neumann function Yν in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions K(x,y) and L(x,y), and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions Vμ,ν(r)(x,y),Wμ,ν(r)(x,y) and the r positive integer of the initial extensions Vμ,ν(x,y),Wμ,ν(x,y). Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions Ψ2(r). Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind Qην in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for Vμ,ν(x,y) and Wμ,ν(x,y). Particularly interesting results are presented for the Neumann function Yν and for the Struve Hν function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article
35 pages, 433 KiB  
Article
Some New Families of Finite Orthogonal Polynomials in Two Variables
by Esra Güldoğan Lekesiz and Iván Area
Axioms 2023, 12(10), 932; https://doi.org/10.3390/axioms12100932 - 29 Sep 2023
Viewed by 978
Abstract
In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family [...] Read more.
In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle’s flavor. Full article
14 pages, 316 KiB  
Article
On a Resolution of Another Isolated Case of a Kummer’s Quadratic Transformation for 2F1
by Mohamed Jalel Atia and Ahmed Saleh Al-Mohaimeed
Axioms 2023, 12(2), 221; https://doi.org/10.3390/axioms12020221 - 20 Feb 2023
Cited by 3 | Viewed by 1538
Abstract
It is well-known that the Kummer quadratic transformation formula is valid provided that its parameters fulfill some specific conditions (see Gradshteyn, Ryzhik, Tables of Integrals, Series and Products, 9.130, 9.134.1). Very recently, one of us established a new identity when one of these [...] Read more.
It is well-known that the Kummer quadratic transformation formula is valid provided that its parameters fulfill some specific conditions (see Gradshteyn, Ryzhik, Tables of Integrals, Series and Products, 9.130, 9.134.1). Very recently, one of us established a new identity when one of these conditions is not fulfilled. In this paper, we aim to discuss another isolated case which completely different from the first. Moreover, in the end, we mention two interesting consequences of these two new results. Full article
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