Discrete inequalities, orthogonal polynomials and the spectral theory of difference operators

1992 ◽  
Vol 437 (1900) ◽  
pp. 355-373 ◽  
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Moment Problem ◽  
Discrete Analogue ◽  
Difference Operators ◽  
Differential Inequalities ◽  
Classical Orthogonal Polynomials ◽  
Best Constant ◽  
Discrete Inequalities ◽  
Three Term Recurrence Relation

In a recent paper the first two authors studied a class of series inequalities associated with a three-term recurrence relation which includes a well-known inequality of Copson’s. It was shown that the validity of the inequality and the value of the best constant are determined in term s of the so-called Hellinger-Nevanlinnam -function. The theory is the discrete analogue of that established by Everitt for a class of integro-differential inequalities. In this paper the properties of the m -function are investigated and connections with the theory of orthogonal polynomials and the H am burger moment problem are explored. The results are applied to give examples of the series inequalities associated with the classical orthogonal polynomials.

scholarly journals Tridiagonal Operators and Zeros of Polynomials in Two Variables

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Chrysi G. Kokologiannaki ◽  
Eugenia N. Petropoulou ◽  
Dimitris Rizos
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Adjoint Operator ◽  
Zeros Of Polynomials ◽  
Analytic Method ◽  
Classical Orthogonal Polynomials ◽  
Three Term Recurrence Relation

The aim of this paper is to connect the zeros of polynomials in two variables with the eigenvalues of a self-adjoint operator. This is done by use of a functional-analytic method. The polynomials in two variables are assumed to satisfy a five-term recurrence relation, similar to the three-term recurrence relation that the classical orthogonal polynomials satisfy.

Generating Some Symmetric Semi-classical Orthogonal Polynomials

2015 ◽  
Vol 58 (4) ◽  
pp. 877-890
Author(s):  
Mohamed Zaatra
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Linear Functional ◽  
Symmetric Form ◽  
Classical Orthogonal Polynomials ◽  
Classical Form ◽  
Knowing That ◽  
Structure Relation ◽  
Three Term Recurrence Relation ◽  
Structure Relation Coefficients

AbstractWe show that if v is a regular semi-classical form (linear functional), then the symmetric form u defined by the relation x2σu = -λv, where (σƒ )(x) = f (x2) and the odd moments of u are 0, is also regular and semi-classical form for every complex λ except for a discrete set of numbers depending on v. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with u and the class of the form u knowing that of v. We conclude with an illustrative example.

scholarly journals LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 10 (24) ◽  
pp. 1795-1802 ◽  
Author(s):  
YURI SMIRNOV ◽  
ALEXANDER TURBINER
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Heisenberg Algebra ◽  
Difference Operators ◽  
Discrete Version ◽  
Classical Orthogonal Polynomials ◽  
Finite Difference Equations ◽  
Exactly Solvable

A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl 2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

Orthogonal polynomials with respect to the form u = λ(x – a)–1 ν + δb

2010 ◽  
Vol 17 (3) ◽  
pp. 581-596
Author(s):  
Mabrouk Sghaier
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Linear Functional ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Regular Form ◽  
Necessary And Sufficient ◽  
Three Term Recurrence Relation

Abstract We study properties of the form (linear functional) u = λ(x – a)–1 ν + δb , where ν is a regular form. We give a necessary and sufficient condition for the regularity of the form u. The coefficients of a three-term recurrence relation, satisfied by the corresponding sequence of orthogonal polynomials, are given explicitly. The semi-classical character of the founded families is studied. We conclude by giving some examples.

scholarly journals Turán inequalities for symmetric orthogonal polynomials

1997 ◽  
Vol 20 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Joaquin Bustoz ◽  
Mourad E. H. Ismail
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Bessel Functions ◽  
Sum Of Squares ◽  
Weighted Sum ◽  
Pollaczek Polynomials ◽  
Lommel Polynomials ◽  
Three Term Recurrence Relation

A method is outlined to express a Turán determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials andq-Bessel functions.

Generating functions and semi-classical orthogonal polynomials

1994 ◽  
Vol 124 (5) ◽  
pp. 1003-1011 ◽  
Author(s):  
Pascal Maroni ◽  
Jeannette Van Iseghem
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Special Form ◽  
Classical Orthogonal Polynomials ◽  
Order Linear ◽  
First Order

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

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