On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Abstract

Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let D ν be the corresponding difference operator defined in the usual way either by(q−1)x . Let U and V be two moment regular linear functionals and let {P n (x)} n≥0 and {Q n (x)} n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two OPS {P n (x)} n≥0 and {Q n (x)} n≥0 assuming that their difference derivatives D ν of higher orders m and k (resp.) are connected by a linear algebraic structure relation such aswhere M, N, m, k ∈ N ∪ {0}, a M,n = 0 for n ≥ M , b N,n = 0 for n ≥ N , and a i,n = b i,n = 0 for i > n. Under certain conditions, we prove that U and V are related by a rational factor (in the ν−distributional sense). Moreover, when m = k then both U and V are D νsemiclassical functionals. This leads us to the concept of (M, N )-D ν -coherent pair of order (m, k) extending to the discrete case several previous works. As an application we consider the OPS with respect to the following Sobolev-type inner productassuming that U and V (which, eventually, may be represented by discrete measures sup-ported either on a uniform lattice if ν = ω, or on a q-lattice if ν = q) constitute a (M, N )-D ν -coherent pair of order m (that is, an (M, N )-D ν -coherent pair of order (m, 0)), m ∈ N being fixed.

(1,1)-Coherent Pairs on the Unit Circle

“…Likewise, Marcellán and Pinzón-Cortés in [11,12] studied the analogue of the generalized coherent pairs introduced by Delgado and Marcellán, that is, (1, 1)--coherent pairs and (1, 1)--coherent pairs. Finally, Alvarez-Nodarse et al [13] analyzed the more general case, ( , )--coherent pairs of order ( , ) and ( , )-coherent pairs of order ( , ), proving the analogue results to those in [4]. Furthermore, Branquinho et al in [14] extended the concept of coherent pair to Hermitian linear functionals associated with nontrivial probability measures supported on the unit circle.…”

“…is a Carathéodory function. Conversely, the Herglotz representation theorem claims that every Carathéodory function ( ) has a representation given by (13) for a unique probability measure on T.…”

Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

“…This leads to ∆-SOP or q-SOP. In many cases the approach to study these SOP has been made considering particular weights or only one of these two difference operators (see, among others, [4,5,6,7,14,21,23,24,29,31]) although there are some papers where the three cases have been treated in a unified way via the Hahn difference operator as, for example, in [1,15].…”

“…We denote by {Q n } n≥0 the sequence of monic orthogonal polynomials with respect to the inner product (1). We also denote by {P n } n≥0 the sequence of monic polynomials orthogonal with respect to the inner product…”

Generalized coherent pairs on the unit circle and Sobolev orthogonal polynomials

“…From this pioneering contribution, coherent pairs and Sobolev orthogonal polynomials have been widely studied and extended in recent decades (for a historical summary, see e.g., the introductory sections in the recent papers [6] and [8] as well as [9]. Lately, de Jesus, Marcellán, Petronilho, and Pinzón-Cortés [5] generalized all those works focussing the attention on the Sobolev inner product Similarly, in the framework of orthogonal polynomials of a discrete variable, Álvarez-Nodarse, Petronilho, Pinzón-Cortés, and Sevinik-Adıgüzel [1] analyzed the case when µ 0 and µ 1 are discrete measures supported either on a uniform lattice or on a q-lattice, and they are an (M, N )-D ν -coherent pair of order m, ν = ω ∈ C {0} or ν = q ∈ C {0, 1}, when in (1.1) instead of the standard derivative operator you consider either D ω p(x) = p(x+ω)−p (x) ω or D q p(x) = p(qx)−p (x) (q−1)x , the difference and the q-difference operator, respectively.…”