d-Orthogonality via generating functions (original) (raw)

SOME BASIC d-ORTHOGONAL POLYNOMIAL SETS

The purpose of this paper is to study the class of polynomial sets which are at the same time d-orthogonal and q-Appell. By a linear change of variable, the resulting set reduces to q-Al-Salam-Carlitz polynomials, for d = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order d + 1. We also explicitly express a d-dimensional functional for which the d-orthogonality holds.

Some discrete d-orthogonal polynomial sets

Journal of Computational and Applied Mathematics, 2003

In this paper, we characterize all polynomial sets which are at the same time d-orthogonal and !-Appell. The resulting polynomials reduce to Charlier polynomials for (d; !)=(1; 1). Various properties of the obtained polynomials are singled out: generating function, recurrence relation of order d + 1 and a di erence equation of order d + 1. We also explicitly express the d-dimensional functional for which the d-orthogonality holds.

On the classical {$d$}-orthogonal polynomials defined by certain generating functions. I

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2000

This paper is a direct sequel to [5]. The present part deals with the problem of finding all d-orthogonal polynomial sets generated by G(x, t) = e t Ψ(xt). The resulting polynomials reduce to Laguerre polynomials for d=1 and to two-orthogonal polynomials associated with MacDonald functions for d=2, recently considered by the authors [6] and by Van Assche and Yakubovich [36]. Various properties for the obtained polynomials are singled out.

d-Orthogonal Analogs of Classical Orthogonal Polynomials

Symmetry, Integrability and Geometry: Methods and Applications, 2018

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.

A generalized hypergeometric d -orthogonal polynomial set

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the context of d-orthogonality. The resulting polynomials are analogous to the classical Laguerre polynomials. We provide some of their properties. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Les polynômes d-orthogonaux de type hypergéométriques généralisés Résumé. On pésente, dans cette Note, une extension du résultat de Abdul-Halim et Al-Salam [1], dans le contexte de la d-orthogonalité. Les polynômes d-orthogonaux obtenus sont analogues aux polynômes classiques de Laguerre. Nous donnons quelques propriétés de cette suite de polynômes. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée (Les numéros d'équations renvoient à la version anglaise.) Les fonctions hypergéométriques généralisées p F q (z) = p F q (a p); (b q); z sont définies par (1.1). Si un des paramètres a p , du numérateur, est un entier négatif ou nul, la série p F q se termine et conduit à un polynôme de la forme hypergéométrique généralisée (1.2). Dans [1], Abdul-Halim et Al-Salam ont traité le problème de la détermination de tous les polynômes orthogonaux de type (1.2). Ils montrent que les seuls polynômes orthogonaux de ce type sont essentiellement les polynômes de Laguerre L (α) n représentés par (1.4). Les suites de polynômes orthogonaux sont caractérisées par le fait qu'elles vérifient une relation de récurrence d'ordre deux. En particulier, les polynômes orthogonaux classiques (Hermite, Laguerre, Bessel, Jacobi) sont les suites orthogonales dont la suite des dérivées est aussi orthogonale (propriété de Hahn [5]). La d-orthogonalité généralise d'une manière naturelle l'orthogonalité habituelle. Elle se présente comme la notion adéquate dans l'étude des récurrences de polynômes d'ordre d + 1. Le résultat fondamental qui Note présentée par Charles-Michel MARLE. S0764-4442(00)01661-X/FLA (α d) n et les polynômes Z α n (x, k) (définis par (3.4)), considérés par Toscano [10] et connus dans la littérature par les polynômes de Konhauser.

Vector Orthogonal Polynomials of Dimension

Vector orthogonal polynomials of dimension ?d where d is a nonzero positive integer are de ned. They are proved to satisfy a recurrence relation with d + 2 terms. A Shohat{Favard type theorem and a QD like algorithm are given.

d-Orthogonal polynomials and

Journal of Mathematical Analysis and Applications, 2012

Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows for their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.). In the limit where su(2) contracts to the Heisenberg-Weyl algebra h 1 , the polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.

New Classes of Orthogonal Polynomials

We show that two new classes of orthogonal polynomials can be derived by applying two orthogonalization procedures due to Löwdin to a set of monomials. They are new in that they possess novel properties in terms of their inner products with the monomials. Each class comprises sets of orthogonal polynomials that satisfy orthogonality conditions with respect to a weight function on a certain interval.

Ond-orthogonal polynomials of Sheffer type

Journal of Difference Equations and Applications, 2018

The polynomial sequences of Sheffer type {P n } n≥0 are defined by the following generating function: ∞ n=0 P n (x) n! t n = A(t)e xC(t). In this work, we are interested, with these sequences when they are also d-orthogonal polynomial sets, that is to say polynomials satisfying one standard (d + 1)-order recurrence relation. We revisit some families in the literature and we state an explicit formula giving the exact number of Sheffer type d-orthogonal sets. We investigate, in details, the d-symmetric case and the particular case d = 3.