D-orthogonal Polynomials Research Articles

D-orthogonality and q-Askey-scheme

Abstract In this work, by solving a d-Geronimus problem we introduce new examples of basic hypergeometric d-orthogonal polynomials which are useful to construct a similar table to the q-Askey-scheme in the context of d-orthogonality. For d = 1, this leads to a characterization theorem involving all polynomials belonging to the q-Askey-scheme, except the continuous q-Hermite ones.

Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra suq(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.

D-Symmetric d-orthogonal polynomials of Brenke type

In this paper, we characterize the d-symmetric d-orthogonal polynomials of Brenke type. We obtain two new families of polynomials and the moments of the corresponding d-dimensional vectors of linear functionals. Weights, providing integral representations for these moments, were also given.

Q-Oscillator Algebra And d-Orthogonal Polynomials

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.

On associated and co-recursive d-orthogonal polynomials

Abstract The 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.

On semi‐classical d‐orthogonal polynomials

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.

Some Inverse Problems for d-Orthogonal Polynomials

This paper is devoted to the study of the generalized inverse problem of the left product of a d–dimensional vector form by a polynomial. The objective is to find the regularity conditions of the vector linear form \({\mathcal{V}}\) defined by \({\mathcal{U} = \mathcal{RV}}\), where \({\mathcal{R}}\) is a d × d matrix polynomial. In such a case, the d–OPS {Qn}n ≥ 0 corresponding to \({\mathcal{V}}\) is d–quasi– orthogonal of order l with respect to \({\mathcal{U}}\). Secondly, we study the inverse problem: Given a d -OPS Pnn ≥ 0 with respect to \({\mathcal{U}}\), characterize the parameters \({\{a^{(i)}_{n}\}{^{dl}_{i=1}}}\) such that the sequence $${Q_{n+dl} = P_{n+dl} + \sum _{i=1}^{dl} a_{n+dl}^{(i)}P_{n+dl-i},\quad n\geq 0}$$ , is d–orthogonal with respect to some regular vector linear form \({\mathcal{V}}\). As an immediate consequence, find the explicit relation between \({\mathcal{U}}\) and \({\mathcal{V}}\).

Formula omitted]-orthogonality of discrete [formula omitted]-Hermite type polynomials

In this paper, we solve a characterization problem involving a suitable basic-hypergeometric form of a polynomial set. That allows us to introduce new examples of Lq-classical d-orthogonal polynomials, generalizing the discrete q-Hermite polynomials in the context of d-orthogonality, and a q-analogous for the d-orthogonal polynomials of Gould–Hopper. For the resulting polynomials, we derive miscellaneous properties. Those turn out to be limit relations, recurrence relations of order (d+1), difference formulas, generating functions, inversion formulas, and d-dimensional functional vectors.

D-Orthogonal polynomials and [formula omitted

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 h1, the polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.

Automorphisms of the Heisenberg–Weyl algebra and d-orthogonal polynomials

We show that the d-orthogonal Charlier and Hermite polynomials appear naturally as matrix elements of nonunitary transformations corresponding to automorphisms of the Heisenberg–Weyl (oscillator) algebra. Basic properties (duality, recurrence, and difference equations) of the d-orthogonal Charlier polynomials can be derived directly from representations of the Heisenberg–Weyl algebra.

On the zeros of d-symmetric d-orthogonal polynomials

In this paper, we give some properties of the zeros of d-symmetric d-orthogonal polynomials and we localize these zeros on ( d + 1 ) rays emanating from the origin. We apply the obtained results to some known polynomials. In particular, we partially solve the conjecture about the zeros of the Humbert polynomials stated by Milovanović and Dordević [G.V. Milovanović, G.B. Dordević, On some properties of Humbert's polynomials, II, Ser. Math. Inform. 6 (1991) 23–30]. A study of the eigenvalues of a particular banded Hessenberg matrix is done.

D-Orthogonality of Humbert and Jacobi type polynomials

In this paper, we treat three questions related to the d-orthogonality of the Humbert polynomials. The first one consists to determinate the explicit expression of the d-dimensional functional vector for which the d-orthogonality holds. The second one is the investigation of the components of Humbert polynomial sequence. That allows us to introduce, as far as we know, new d-orthogonal polynomials generalizing the classical Jacobi ones. The third one consists to solve a characterization problem related to a generalized hypergeometric representation of the Humbert polynomials.

Dunkl–Appell d-orthogonal polynomials

In this paper, we characterize the Dunkl–Appell d-orthogonal polynomials. Various properties of the obtained polynomials are singled out: a generating function, a (d+1)-order recurrence relation and a differential-difference equation. We also explicitly express the d-dimensional functional for which the d-orthogonality holds. A detailed discussion of the (d+1)-symmetric solutions is presented. Some relationships between the obtained polynomials and some known orthogonal polynomial families are established.

On d-orthogonality of the Sheffer systems associated to a convolution semigroup

In this note we investigate which Sheffer polynomials can be associated to a convolution semigroup of probability measures, usually induced by a stochastic process with stationary and independent increments. From a recent kind of d-orthogonality ( d ∈ { 2 , 3 , … } ), we characterize the associated d-orthogonal polynomials by the class of generating probability measures, which belongs to the natural exponential family with polynomial variance functions of exact degree 2 d - 1 . This extends some results of (classical) orthogonality; in particular, some new sets of martingales are then pointed out. For each integer d ⩾ 2 we completely illustrate polynomials with ( 2 d - 1 )-term recurrence relation for the families of positive stable processes.

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

The purpose of this work is to present some results on the d-orthogonal polynomials dened by generating functions of certain forms to be specied below. The resulting polynomials are natural extensions of some classical orthogonal polynomials. The rst part of this study is motived by the recent work of Von Bachhaus [21] who showed that, among the orthogonal polynomials, only the Hermite and the Gegenbauer polynomials are dened by the generating function G 2xt t 2 . Here we generalize this result in the context of d-orthogonality, by considering the polynomials generated by G (d +1)xt t d+1 ,w hered is a positive integer. We obtain that the resulting polynomials are d-symmetric Denition 1.2 and \classical in the Hahn’s sense. We provide some examples to illustrate the results obtained and show that they involve certain known polynomials. Finally, we conclude by giving some properties of the zeros of these polynomials as well as a (d +1 )-order dierential equation satised by each polynomial. In forthcoming paper [2] we will consider the polynomials generated by e t (xt).

The relation of the d-orthogonal polynomials to the Appell polynomials

We are dealing with the concept of d-dimensional orthogonal ( abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials. A sequence of these polynomials is obtained. All the elements of its ( d + 1)-order recurrence are explicitly determined. A generating function, a ( d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.

A Characterization of "Classical" d-Orthogonal Polynomials

We give a characterization of "classical" d-orthogonal polynomials through a vectorial functional equation. A sequence of monic polynomials { B n } n ≥ 0 is called d-simultaneous orthogonal or simply d-orthogonal if it fulfils the following d + 1-st order recurrence relation: [formula] with the initial conditions [formula] Denoting by { L n } n ≥ 0 the dual sequence of { B n } n ≥ 0 , defined by 〈 L n , B m 〉 = δ n, m, n, m ≥ 0, then the sequence { B n } n ≥ 0 is d-orthogonal if and only if [formula] for any integer α with 0 ≤ α ≤ d − 1. Now, the d-orthogonal sequence { B n } n ≥ 0 is called "classical" if it satisfies the Hahn′s property, that is, the sequence { Q n } n ≥ 0 is also d-orthogonal where Q n ( x) = ( n + 1) − 1 B′ n + 1 ( x), n ≥ 0 is the monic derivative. If Λ denotes the vector t ( L 0, L 1, ..., L d − 1 ), the main result is the following: the d-orthogonal sequence { B n } n ≥ 0 is "classical" if and only if, there exist two d × d polynomial matrices Ψ = (ψ ν, μ), Φ = (φ ν, μ), deg ψ ν, μ ≤ 1, deg φ ν, μ ≤ 2 such that Ψ Λ + D(Ψ Λ) = 0 with conditions about regularity (see below). Moreover, some examples are given.