About families of orthogonal polynomials satisfying Heun’s differential equation

2021 ◽  
Vol 263 ◽  
pp. 105522
Author(s):  
Alphonse P. Magnus ◽  
François Ndayiragije ◽  
André Ronveaux
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

scholarly journals The second-order self-associated orthogonal sequences

2004 ◽  
Vol 2004 (2) ◽  
pp. 137-167 ◽  
Author(s):  
P. Maroni ◽  
M. Ihsen Tounsi
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Second Order ◽  
Integral Representations ◽  
Second Order Differential Equation ◽  
Structure Relation ◽  
Orthogonal Sequences

The aim of this work is to describe the orthogonal polynomials sequences which are identical to their second associated sequence. The resulting polynomials are semiclassical of classs≤3. The characteristic elements of the structure relation and of the second-order differential equation are given explicitly. Integral representations of the corresponding forms are also given. A striking particular case is the case of the so-called electrospheric polynomials.

Differential Equation for Classical-Type Orthogonal Polynomials

1989 ◽  
Vol 32 (4) ◽  
pp. 404-411 ◽  
Author(s):  
A. Ronveaux ◽  
F. Marcellan
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Arbitrary Number ◽  
Order Differential Equation ◽  
Second Order ◽  
Classical Type ◽  
Dirac Mass ◽  
Second Order Differential Equation

AbstractThe second order differential equation of Littlejohn-Shore for Laguerre type orthogonal polynomials is generalized in two ways. First the positive Dirac mass can be situated at any point and secondly the weight can be any classical weight modified by an arbitrary number of Dirac distributions.

Ladder operators and a differential equation for varying generalized Freud-type orthogonal polynomials

2018 ◽  
Vol 07 (04) ◽  
pp. 1840005 ◽  
Author(s):  
Galina Filipuk ◽  
Juan F. Mañas-Mañas ◽  
Juan J. Moreno-Balcázar
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Inner Product ◽  
Ladder Operators ◽  
First Order ◽  
Standard Family ◽  
Differential Difference Equation ◽  
First Order Differential Equations

In this paper, we introduce varying generalized Freud-type polynomials which are orthogonal with respect to a varying discrete Freud-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as find a second-order differential–difference equation that these polynomials satisfy. To reach this objective, it is necessary to consider the standard Freud orthogonal polynomials and, in the meanwhile, we find new difference relations for the coefficients in the first-order differential equations that this standard family satisfies.

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1)

1984 ◽  
Vol 27 (2) ◽  
pp. 205-214 ◽  
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Differential Equation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Hypergeometric Functions ◽  
Order Differential Equation ◽  
Second Order Differential Equations ◽  
Special Cases ◽  
Delta Functions ◽  
Fourth Order Differential Equation

AbstractWe study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

scholarly journals An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

2020 ◽  
pp. 86-96
Author(s):  
Galina A. Rasolko ◽  
Sergei M. Sheshko
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Electromagnetic Waves ◽  
Logarithmic Singularity ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Solution Function ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed.  This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative.  The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals Orthogonal polynomials for the oscillatory-Gegenbauer weight

2008 ◽  
Vol 84 (98) ◽  
pp. 49-60 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic ◽  
Zvezdan Marjanovic
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Existence Theorem ◽  
Linear Space ◽  
Linear Functional ◽  
Order Differential Equation ◽  
Algebraic Polynomials ◽  
Recurrence Coefficients ◽  
Differential Relations ◽  
Previous Existence

This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P?C, where L = ?1 -1 p(x) d?(x), d?(x) = (1-x?)?-1/2 exp(i?x) dx, and P is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational ? ? (-1/2,0], give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation.

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