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43 pages, 521 KiB

Open AccessArticle

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 237

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

17 pages, 1187 KiB

Open AccessArticle

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 414

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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19 pages, 982 KiB

Open AccessArticle

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 378

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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10 pages, 273 KiB

Open AccessArticle

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 511

Abstract

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials

{K_{n}}_{n = 0}^{\infty}

—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space

H^{1} [- 1, 1]

with respect to [...] Read more.

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials

{K_{n}}_{n = 0}^{\infty}

—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space

H^{1} [- 1, 1]

with respect to the (positive-definite) inner product

{(f, g)}_{1, c} : = - \frac{(f (1) - f (- 1)) (\bar{g} (1) - \bar{g} (- 1))}{2} + \int_{- 1}^{1} (f^{'} (x) {\bar{g}}^{'} (x) + c f (x) \bar{g} (x)) d x,

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

K_{c}

(c > 0)

in

L^{2} (- 1, 1) .

Other than

K_{0}

and

K_{1},

these polynomials are not eigenfunctions of

K_{c} .

As shown by Littlejohn and Quintero, the sequence

{K_{n}}_{n = 0}^{\infty}

forms a complete orthogonal set in the first left-definite space

(H^{1} [- 1, 1], {(\cdot, \cdot)}_{1, c})

associated with

(K_{c},

L^{2} (- 1, 1))

. Furthermore, they show that, for

n \geq 1,

K_{n} (x)

has n distinct zeros in

(- 1, 1) .

In this note, we find an explicit formula for Krein–Sobolev polynomials

{K_{n}}_{n = 0}^{\infty}

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

33 pages, 737 KiB

Open AccessFeature PaperArticle

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 700

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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12 pages, 270 KiB

Open AccessArticle

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 770

Abstract

In 1812, Gauss stated the following identity: [...] Read more.

In 1812, Gauss stated the following identity:

{}_{2}F_{1} (a, b; c; 1) = \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)}

, where, in the real case,

c - a - b > 0

and as an immediat consequence the Chu–Vandermonde identity:

{}_{2}F_{1} (a, - n; c; 1) = \frac{{(c - a)}_{n}}{{(c)}_{n}}

for any positive integer n. In this paper, we investigate the case when

c - a - b < 0

by taking

c = 2 b = - 2 n

, n and a are positive integers (

c - a - b = - n - a < 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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

20 pages, 465 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

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 1348

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

(D^{α} f) (y) = \frac{1}{Γ (m - α)} \int_{0}^{y} {(y - x)}^{m - α - 1} f^{(m)} (x) d x, y > 0

, with

m - 1 < α \leq m, m \in N .

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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17 pages, 354 KiB

Open AccessArticle

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 870

Abstract

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

in [...] Read more.

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

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

V_{μ, ν}^{(r)} (x, y), W_{μ, ν}^{(r)} (x, y)

and the r positive integer of the initial extensions

V_{μ, ν} (x, y), W_{μ, ν} (x, y)

. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions

Ψ_{2}^{(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

Q_{η}^{- ν}

in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for

V_{μ, ν} (x, y)

and

W_{μ, ν} (x, y)

. Particularly interesting results are presented for the Neumann function

Y_{ν}

and for the Struve

H_{ν}

function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

35 pages, 433 KiB

Open AccessArticle

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 1002

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

14 pages, 316 KiB

Open AccessArticle

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 1559

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

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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