scholarly journals Fourt-order differential equation satisfied by the associated of any order of all classical orthogonal polynomials. A study of their distribution of zeros

1993 ◽  
Vol 49 (1-3) ◽  
pp. 349-359 ◽  
Author(s):  
A. Zarzo ◽  
A. Ronveaux ◽  
E. Godoy
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Distribution Of Zeros ◽  
Classical Orthogonal Polynomials

scholarly journals Some properties of orthogonal polynomials satisfying fourth order differential equations

1995 ◽  
Vol 37 (1) ◽  
pp. 105-113 ◽  
Author(s):  
R. G. Campos ◽  
L. A. Avila
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Polynomial Solution ◽  
Second Order ◽  
System Of Nonlinear Equations ◽  
Singular Boundary ◽  
Classical Orthogonal Polynomials ◽  
Second Order Differential Equation

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

Orthogonal polynomials satisfying fourth order differential equations

1981 ◽  
Vol 87 (3-4) ◽  
pp. 271-288 ◽  
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Jacobi Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
Absolutely Continuous ◽  
Independent Variable ◽  
Fourth Order Differential Equation

SynopsisThese polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.

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.

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.

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