D-orthogonal Polynomials Research Articles

Multiple orthogonal polynomials, 𝑑-orthogonal polynomials, production matrices, and branched continued fractions

I analyze an unexpected connection between multiple orthogonal polynomials, d d -orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.

On d-orthogonal polynomials with Brenke type generating functions

The polynomial sequences of Brenke type {Pn}n≥0 are defined by the following generating function:∑n=0∞Pn(x)n!tn=A(t)B(xt), where A and B are two formal power series subject to the conditions A(0)B(k)(0)≠0,k=0,1,2….In this work, we characterize all d-orthogonal polynomial sets of Brenke type. That allows us to obtain several new and known results. We give some examples as illustration.

Operational rules for a new family of d-orthogonal polynomials of Laguerre type

The aim of this research is to present a new generalization of d-orthogonal polynomials of Laguerre type by utilizing a suitable generating function from Sheffer class and employing operational rules associated with the lowering and raising operators that satisfy d-orthogonality. We derive several properties of these polynomials and establish the recurrence relation. Moreover, we provide explicit, connection and inversion formulas, the -order differential equation, and the canonical d-dimensional functional vector.

Approximation by Szasz–Kantorovich type operators associated with d-symmetric d-orthogonal polynomials of Brenke type

Approximation by Szasz–Kantorovich type operators associated with d-symmetric d-orthogonal polynomials of Brenke type

On connection between zeros and d-orthogonality

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials is done by K. Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence , with the existence of a certain ratio expressed by means of the zeros of . Then, for the -fold symmetric polynomials, , such that has distinct positive real zeros, , we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio . Finally, we give necessary and sufficient conditions on the zeros of a given sequence , that will assure that this sequence satisfies a particular -order recurrence relation. Many examples to illustrate the obtained results are given.

On the quasi-orthogonality and Hahn-classical d-orthogonal polynomials

This paper focuses on the quasi-orthogonality between sequences of -orthogonal polynomials and their characterization in terms of linear combinations as well as the left product of vector of linear forms by matrix polynomial. Therefore, structure relations for Hahn-classical -orthogonal polynomials are provided.

Operational rules and d-orthogonal polynomials of Laguerre type

The main aim of this paper is to introduce and investigate a new d-orthogonal family generalizing the Laguerre polynomial set. We use operational rules associated with corresponding lowering, raising and transfer operators to derive many related properties, namely, recurrence relation, d-orthogonality, functional vector of d-orthogonality, inversion and connection relations, differential equation, among others.

On the zeros of some d-orthogonal polynomials

In this paper, we give locations of the zeros of polynomials satisfying particular -order recurrence relations and we establish some representations of the smallest and largest real zeros of these polynomials in terms of the parameters in the -order recurrence relation. Particular interest is devoted to -fold symmetric polynomials.

Limit relations involving 2-orthogonal polynomials

ABSTRACT As an extension of the Askey-scheme, recently, it has been introduced the so-called d-Askey-scheme, which is a table describing families of hypergeometric d-orthogonal polynomials. In this paper, we derive limit relations linking between all levels of the 2-Askey-scheme (Figure 1). To this end, we use three methods. The first one is based on a recurrence relation. The second one uses difference and differential equations, and the third one deals with the hypergeometric expression of the considered polynomials. The recurrence relations, differential and difference equations will be determined by the use of Computer Algebra program Maple. That allows us to get, explicitly, the 2-Askey-scheme with various properties of their polynomials, which appears to be new.

Cλ-Extended Oscillator Algebra and d-Orthogonal Polynomials

In this paper we first construct an analytic realization of the Cλ-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of d-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the Cλ-extended oscillator by block matrices.

Discrete Hahn-classical d-orthogonal polynomials

In this paper, we characterize the discrete Hahn-classical d-orthogonal polynomial sets (and their iterated) by a difference distributional matrix equation satisfied by the associated d-dimensional functional vector, where the coefficient matrices are explicitly given. As a result we describe this set of polynomials via a structure relation. We illustrate the preceding results with some examples.

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d=2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

Asymptotic relations involving 𝑑-orthogonal polynomials

Abstract In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

On Mittag–Leffler d-Orthogonal Polynomials

This paper presents a first result of a long term research project dealing with the construction of d-orthogonal polynomials with Hahn's property. We shall show that the latter class could be characterized by expanding a polynomial as a finite sum of first derivatives of the elements of the sequence and we shall explain how this characterization could be used to construct Hahn-classical d-orthogonal polynomials as well. In this paper we look for solutions of linear combinations of the first derivatives of two consecutive elements of the sequence by considering the derivative operator and Delta (discrete) operator. The resulting polynomials constitute a particular class of Laguerre d-orthogonal polynomials and a generalization of Mittag-Leffler polynomials, respectively.

Geronimus transformations for sequences of d-orthogonal polynomials

In this paper an extension of the concept of Geronimus transformation for sequences of $d$-orthogonal polynomials $\{P_n\}$ is introduced. The transformed sequences $\{P^{(k)}_n\}$, for $ k=1,\ldots, d,$ are analyzed and some relationships between these new sequences of polynomials are given. Also the associated Hessenberg matrices and their transformed are studied.

On d-orthogonal polynomials of Sheffer type

ABSTRACTThe polynomial sequences of Sheffer type are defined by the following generating function: 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 -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.

Linear Combinations of d-Orthogonal Polynomials

The regularity of a linear combination of r consecutive elements of a monic d-orthogonal polynomial sequence has been studied when $$(r-1)$$ is a multiple of d. This problem can be solved without the introduction of the concept of d-quasi-orthogonality. In the present contribution, we deal with the problem of d-orthogonality of a linear combination of two elements of a monic d-orthogonal polynomial sequence. This problem is also equivalent to the study the regularity of the left product of a linear form by a matrix polynomial of degree one.

Connection and inversion coefficients for basic hypergeometric polynomials

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.

English

This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1. The underlying algebraic framework allowed a systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy.

Riordan arrays and d-orthogonality

Using the theory of the Riordan group and d-orthogonal polynomials, we shall show that sequences of d-orthogonal polynomials, can be also generated by the Riordan group. We interpret some families of d-orthogonal polynomials within the framework of the Riordan group.