Characterization of the Dunkl-classical symmetric orthogonal polynomials

2007 ◽  
Vol 187 (1) ◽  
pp. 105-114 ◽  
Author(s):  
Y. Ben Cheikh ◽  
M. Gaied
Keyword(s):  
Orthogonal Polynomials

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.

A characterization of some q-orthogonal polynomials

2006 ◽  
Vol 12 (3) ◽  
pp. 425-437 ◽  
Author(s):  
Somjit Datta ◽  
James Griffin
Keyword(s):  
Orthogonal Polynomials

d-dimensional orthogonal polynomials: Commutator theorem and other results

2017 ◽  
Vol 20 (02) ◽  
pp. 1750011
Author(s):  
Luigi Accardi ◽  
Abdallah Dhahri
Keyword(s):  
Orthogonal Polynomials ◽  
Random Variable ◽  
Product Measure ◽  
Classical Formula ◽  
Variable Formula ◽  
Commutator Theorem ◽  
The Creation ◽  
New Formulation ◽  
Preservation Operator

We give new and simplified proofs of three basic theorems in the theory of orthogonal polynomials associated to a classical, [Formula: see text]-valued random variable [Formula: see text] with all moments, namely: (1) The characterization of [Formula: see text] in terms of commutators among the creation–annihilation–preservation (CAP) operators in its quantum decomposition. (2) The characterization, in terms of the same objects, of the fact that the distribution of [Formula: see text] is a product measure. (3) The equivalence of the symmetry of [Formula: see text] with the vanishing of the associated preservation operator. Our new formulation of these results allows one to obtain a stronger form of the above statements.

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 GENERATING FUNCTIONS OF CAUCHY–STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (01) ◽  
pp. 91-98 ◽  
Author(s):  
MAREK BOŻEJKO ◽  
NIZAR DEMNI
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Short Proof ◽  
Probabilistic Interpretation ◽  
Infinitely Divisible ◽  
The Family ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Stieltjes Type

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

Sign in / Sign up

Export Citation Format

Share Document