(1,1)-D_ω-coherent pairs

Abstract

In this work, we introduce the notion of (1, 1)-Dω-coherent pair of weakly quasi-definite linear functionals (U, V) as the Dω-analogue to the generalized coherent pair studied by A. Delgado and F. Marcellán in [8]. This means that their corresponding families of monic orthogonal poly-We prove that (1, 1)-Dω-coherence is a sufficient condition for the weakly quasi-definite linear functionals to be Dω-semiclassical, one of them of class at most 1 and the another of class at most 5, and they are related by a expression of rational type. Additionally, a matrix interpretation of (1, 1)-Dω-coherence in terms of the corresponding monic Jacobi matrices is given. The particular case when U is Dω-classical linear functional is studied.

On linearly related sequences of difference derivatives of discrete orthogonal polynomials

“…In the next theorems, we state the D ν -analogue results obtained in [7,8,14], and we generalize the results stated in [2,4,12,15] for ν = ω, and in [3,16] for ν = q, respectively. Moreover, we give a complete description of the D ν -semiclassical discrete orthogonal polynomials in the framework of (M, N )-D ν -coherence of order (m, k).…”

“…Area, E. Godoy, and F. Marcellán, where these authors present very interesting examples, giving the classification of all q−coherent pairs of positive-definite linear functionals when one of them is either the little q−Jacobi linear functional or the little q−Laguerre linear functional. With this respect see also the more recent work [16].…”

“…One of the most important problems in the theory of Sobolev OPS is to find the explicit expressions for the polynomials themselves. When the Sobolev OPS is orthogonal with respect to the inner product (4.1) and U and V constitute a (M, N )-D ν -coherent pair of order m, it is possible to obtain the Sobolev orthogonal polynomials by using the following two theorems that generalize an algebraic property proved for (M, 0)-D ω -coherent and (1, 1)-D ν -coherent pairs of order 1, in [12,15,16], to (M, N )-D ν -coherent pairs of order m, and they are the D ν -analogue results obtained in [7,9]. Theorem 4.3.…”

“…The concept of coherent pair was extended to the OPS of a discrete variable by I. Area, E. Godoy, and F. Marcellán [2,3,4], and also by F. Marcellán and N. Pinzón-Cortéz [15,16].…”

“…Marcellán and N. C. Pinzón-Cortés ( [15] for ν = ω, [16] for ν = q) showed that if (U, V) is a (1, 1)-D ν -coherent pair then they are D ν -semiclassical linear functionals (one of class at most 1 and the other of class at most 5) and they are related by σ(x)U = ρ(x)V, with deg(σ(x)) ≤ 3, deg(ρ(x)) = 1. Also, they studied the case when U is D ν -classical.…”