scholarly journals Second-order recurrence relation for the linearization coefficients of the classical orthogonal polynomials

1996 ◽  
Vol 69 (1) ◽  
pp. 159-170 ◽  
Author(s):  
Stanisław Lewanowicz
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Second Order ◽  
Classical Orthogonal Polynomials ◽  
Linearization Coefficients

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.

Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Thomas Stoll ◽  
Robert Tichy
Keyword(s):  
Orthogonal Polynomials ◽  
Liouville Equation ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Second Order ◽  
Recurrence Equation ◽  
Classical Orthogonal Polynomials ◽  
The Third ◽  
Sturm Liouville

AbstractIt is well-known that Morgan-Voyce polynomials B n(x) and b n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $$ with $$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $$ with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > −¼B 2 in the third case. We show that the Diophantine equation with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

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.

scholarly journals On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 246 ◽  
Author(s):  
Lino G. Garza ◽  
Luis E. Garza ◽  
Edmundo J. Huertas
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Recurrence Relation ◽  
Second Order ◽  
Connection Formula ◽  
Second Order Differential Equations ◽  
Spectral Transformations ◽  
Structure Formula ◽  
Coherent Pairs

In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle.

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.

scholarly journals Stable Calculation of Krawtchouk Functions from Triplet Relations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1972
Author(s):  
Albertus C. den Brinker
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Error Propagation ◽  
Orthonormal Basis ◽  
Theoretical Background ◽  
Classical Orthogonal Polynomials ◽  
Krawtchouk Polynomials ◽  
Unit Norm

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.

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.

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.

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