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.

The Krall Polynomials as Solutions to a Second Order Differential Equation

1983 ◽  
Vol 26 (4) ◽  
pp. 410-417 ◽  
Author(s):  
Lance L. Littlejohn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Polynomial Solutions ◽  
Second Order Differential Equations ◽  
Image Position

AbstractA popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form1.1

Conjugacy of half-linear second-order differential equations

2000 ◽  
Vol 130 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Ondřej Došlý ◽  
Árpád Elbert
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Focal Point ◽  
Second Order ◽  
Oscillation Criterion ◽  
Riccati Transformation ◽  
Second Order Differential Equations ◽  
Generalized Riccati Transformation ◽  
Image Position

Focal point and conjugacy criteria for the half-linear second-order differential equation are obtained using the generalized Riccati transformation. An oscillation criterion is given in case when the function c(t) is periodic.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya

1972 ◽  
Vol 69 (4) ◽  
pp. 295-333 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

scholarly journals Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng ◽  
Xinqin Pang
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Generalized Variational Principle ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Second Order Differential Equations ◽  
Mixed Nonlinearities

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.

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.

Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian

Analysis ◽  
2017 ◽  
Vol 37 (1) ◽  
pp. 1-11
Author(s):  
Hairong Lian ◽  
Dongli Wang ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Second Order ◽  
Periodic Boundary Value Problem ◽  
Second Order Differential Equations

AbstractIn this paper, we study a periodic boundary value problem for a nonautonomous second-order differential equation with a

scholarly journals Asymptotic integration of the second order differential equation, resonance effect

2015 ◽  
Vol 63 (1) ◽  
pp. 223-235
Author(s):  
Barbara Pietruczuk
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Resonance Effect ◽  
Second Order ◽  
Asymptotic Integration ◽  
Asymptotic Formulas ◽  
Von Neumann ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation

Abstract There will be presented asymptotic formulas for solutions of the equation y'' + (1 + φ (x))y = 0, 0 < x0 < x < ∞ , where function is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function φ . The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner von Neumann potential. A class of potentials generalizing that of Wigner von Neumann will be presented.

scholarly journals On a second order differential equation with piecewise constant mixed arguments

2011 ◽  
Vol 27 (1) ◽  
pp. 1-12
Author(s):  
H. BEREKETOGLU ◽  
◽  
G. SEYHAN ◽  
F. KARAKOC ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Zero Solution ◽  
Second Order ◽  
Uniqueness Of Solutions ◽  
Piecewise Constant ◽  
Second Order Differential Equations ◽  
Mixed Arguments

We prove the existence and uniqueness of solutions of a class of second order differential equations with piecewise constant mixed arguments and we show that the zero solution of Eq. (1.1) is a global attractor. Also, we study some properties of solutions of Eq. (1.1) such as oscillation, nonoscillation, and periodicity.

scholarly journals Local convexity for second order differential equations on a Lie algebroid

2021 ◽  
Vol 13 (3) ◽  
pp. 477
Author(s):  
Juan Carlos Marrero ◽  
David Martín de Diego ◽  
Eduardo Martínez
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Lie Algebroid ◽  
Local Convexity ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation

<p style='text-indent:20px;'>A theory of local convexity for a second order differential equation (${\text{sode}}$) on a Lie algebroid is developed. The particular case when the ${\text{sode}}$ is homogeneous quadratic is extensively discussed.</p>

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