q-Coherent pairs and q-orthogonal polynomials
Introduction
The concepts of coherent pair and symmetric coherent pair have been introduced by Iserles et al. in [1] in the framework of the study of orthogonal polynomials associated with the Sobolev inner productwhere μ0 and μ1 are non-atomic positive Borel measures on the real line such that
In fact, coherence means that a relation between the monic orthogonal polynomial sequence (MOPS) {Pn}n and {Tn}n, associated with the measures μ0 and μ1, respectively,where {σn}n is a sequence of non-zero complex numbers, is satisfied.
Hahn [2] seems to have been the first to realize that the characterizations of classical orthogonal polynomial sequences based on derivatives and differential equations are too much restrictive [3]. He used a more general operator, the so-called q-difference operator defined by [2, Eq. (2.3)]and (Dqf)(0):=f′(0) by continuity, provided f′(0) exists. Note that limq↑1(Dqf)(x)=f′(x) if f is a differentiable function.
The Askey tableau of hypergeometric orthogonal polynomials contains the classical orthogonal polynomials which can be written in terms of a hypergeometric function, starting at the top with Wilson and Racah polynomials and ending at the bottom with Hermite polynomials [4], [5]. Hahn [2] studied the q-analogue of this classification. So, there are q-analogues of all the families in the Askey tableau, often several q-analogues for one classical family. The most general sets of these q-analogues are the Askey–Wilson polynomials [4] and the q-Racah polynomials [6], which contain all other families as special or limit cases [7]. In [8, p. 115] Koornwinder presented a q-Hahn tableau: a q-analogue of that part of the Askey tableau which is dominated by Hahn polynomials. Basic hypergeometric functions and q-orthogonal polynomials for arbitrary (including complex) values of q are connected with quantum algebras and groups [9].
The aim of this paper is to extend the recent study on coherent pairs of linear functionals [10] and Δ-coherent pairs of linear functionals [11], [12] to q-coherent pairs. More concretely, we characterize the sequences of orthogonal polynomials {Pn}n and {Tn}n such thatwhere {σn}n is a sequence of non-zero complex numbers and(see [2, p. 5]). Moreover, we determine all q-coherent pairs of linear functionals when dealing with little q-Jacobi and little q-Laguerre/Wall linear functionals. By using limit properties for linear functionals the classification given by Meijer in the continuous case [10] is reached. In this way, an interesting direction of research can be open. If is a q-coherent pair of positive measures, then the study of q-Sobolev orthogonal polynomials seems to be very natural. A particular case of q-coherent pairs has been developed in [13]. On the other hand, in the Doctoral Dissertation by Koekoek [14] and a subsequent paper [15] it was studied a Sobolev type inner productas a generalization of a q-analogue of the classical Laguerre polynomials.
The outline of the paper is as follows. In Section 2, we give basic definitions and results which will be helpful in the following sections. In Section 3, we present the q-classical linear functionals. In Section 4, we introduce the concept of q-coherent pair of linear functionals, and we prove that if is a q-coherent pair, then both and are q-semiclassical linear functionals. In Section 5, we prove that if is a q-coherent pair of linear functionals, then at least one of them must be a q-classical linear functional. In Section 6, we give the classification of all q-coherent pairs of positive-definite linear functionals when or is either the little q-Jacobi linear functional or the little q-Laguerre linear functional. Finally, in Section 7, by using limit relations we recover the classification of all coherent pairs of positive definite linear functionals.
Section snippets
Notations and basic results
Let be the linear space of complex polynomials and let be its algebraic dual space. We denote by the duality bracket for and , and we denote by , with n⩾0, the canonical moments of . Definition 1 A linear functional is said to be quasi-definite if all the principal submatrices of the infinite Hankel matrix are non-singular.
It is known [16] that a linear functional is quasi-definite if and only if there exists an MOPS {Pn}n orthogonal with respect to ,
q-Classical linear functionals
Definition 7 A functional is said to be a q-classical linear functional if is quasi-definite and there exist polynomials φ and ψ with deg(φ)⩽2 and deg(ψ)=1 such that
The corresponding MOPS associated with is said to be a q-classical MOPS.
In [2], [8] we can find the families of q-classical polynomial sequences namely: Big q-Jacobi, little q-Jacobi, big q-Laguerre, q-Meixner, alternative q-Charlier, little q-Laguerre/Wall, Moak, Al-Salam-Carlitz I, Al-Salam-Carlitz II, Stieltjes-Wigert,
q-Coherent pairs
Let us introduce the concept of q-coherent pair of linear functionals, as a q-analogue of coherent pair of linear functionals, i.e., when q→1 we recover the concept of coherent pair of linear functionals used in [10], [21], [22]. Definition 10 Let and be two quasi-definite linear functionals, whose MOPS are {Pn}n and {Tn}n, respectively. We define as a q-coherent pair of linear functionals ifwhere {σn}n is a sequence of non-zero complex numbers and the
General problem of q-coherence
In Theorem 1 we have proved that if is a q-coherent pair of linear functionals, then both , are q-semiclassical functionals of class at most 6 and 1, respectively. The main goal of this section is to prove that if is a q-coherent pair of linear functionals, then at least one of the functionals , has to be a q-classical functional. In order to give a scheme of the proof let us denote by ξ and η the zeros of the polynomial B2 defined in (26). The proof of this statement
Examples
In Section 5 we have proved that if is a q-coherent pair of linear functionals, at least one of them has to be a q-classical linear functional. In this section we give the coherent pairs of positive-definite linear functionals when one of the functionals is the little q-Jacobi linear functional defined as [23, Eq. (3.12.2)]where the q-shifted factorials (c;q)k are defined in (15) and when one of the linear forms is
The limit transitions
In [10], Meijer proved that if is a coherent pair of linear functionals (2), then at least one of the functionals has to be a classical continuous one, i.e., Hermite, Laguerre or Jacobi linear functional. He gave the classification of coherent pairs of linear functionals which can be represented by distribution functions. In this section, limit transitions from little q-Jacobi linear functional to Jacobi linear functional and from little q-Laguerre linear functional to Laguerre linear
Acknowledgements
E. Godoy wishes to acknowledge partial financial support by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB-96-0952. The research of F. Marcellán has been partially supported by DGES of Spain under Grant PB96-0120-C03-01 and INTAS Project 2000-272.
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