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.

On associated and co-recursive d-orthogonal polynomials

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
A. Sadek Saib ◽  
Ebtissem Zerouki
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Explicit Form ◽  
Stieltjes Function ◽  
Stieltjes Functions ◽  
Definition Of ◽  
Dual Sequence

AbstractThe associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given.

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 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.

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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