scholarly journals ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Nazeer Ahmed Khoso
Keyword(s):  
Orthogonal Polynomials ◽  
Polynomial Sequences ◽  
Classical Polynomials ◽  
Boubaker Polynomials ◽  
Schmidt Orthogonalization ◽  
Sturm Liouville

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

scholarly journals Three-fold symmetric Hahn-classical multiple orthogonal polynomials

2019 ◽  
Vol 18 (02) ◽  
pp. 271-332 ◽  
Author(s):  
Ana F. Loureiro ◽  
Walter Van Assche
Keyword(s):  
Asymptotic Behavior ◽  
Orthogonal Polynomials ◽  
Symmetric Polynomial ◽  
Recurrence Coefficients ◽  
Absolute Value ◽  
Polynomial Sequences ◽  
Full Characterization ◽  
Classical Polynomials ◽  
Multiple Orthogonal Polynomials

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

scholarly journals Exceptional Legendre Polynomials and Confluent Darboux Transformations

2021 ◽  
Author(s):  
María Ángeles García-Ferrero ◽  
◽  
David Gómez-Ullate ◽  
Robert Milson ◽  
◽  
...  
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomials ◽  
Eigenvalue Problems ◽  
Jacobi Polynomials ◽  
The Novel ◽  
Darboux Transformations ◽  
Polynomial Sequences ◽  
New Construction ◽  
Sturm Liouville ◽  
Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

scholarly journals On the analytic form of the discrete Kramer sampling theorem

2001 ◽  
Vol 25 (11) ◽  
pp. 709-715 ◽  
Author(s):  
Antonio G. García ◽  
Miguel A. Hernández-Medina ◽  
María J. Muñoz-Bouzo
Keyword(s):  
Orthogonal Polynomials ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Analytic Form ◽  
Sampling Theorem ◽  
Discrete Version ◽  
Moment Problems ◽  
Sturm Liouville ◽  
The Subject ◽  
Sampling Expansions

The classical Kramer sampling theorem is, in the subject of self-adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm-Liouville problems. In this paper a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

scholarly journals A characterization of the four Chebyshev orthogonal families

2005 ◽  
Vol 2005 (13) ◽  
pp. 2071-2079 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. Garcia-Amor
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Linear Combinations ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Sequences ◽  
Fixed Length ◽  
Constant Coefficients ◽  
Fourth Kind

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

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 Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

2020 ◽  
Vol 10 (02) ◽  
pp. 2050003
Author(s):  
Diego Dominici
Keyword(s):  
Power Series ◽  
Asymptotic Analysis ◽  
Orthogonal Polynomials ◽  
Asymptotic Expansions ◽  
Linear Functional ◽  
Series Expansions ◽  
Recurrence Coefficients ◽  
Variable Formula ◽  
Polynomial Sequences ◽  
Power Series Expansions

We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

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