q-Coherent pairs and q-orthogonal polynomials

Abstract

In this paper we introduce the concept of q coherent pair of linear functionals. We prove that if ðu 0 ; u 1 Þ is a q coherent pair of linear functionals, then at least one of them has to be a q classical linear functional. Moreover, we present the classification of all q coherent pairs of positive definite linear functionals when u 0 or u 1 is either the little q Jacobi linear functional or the little q Laguerre/Wall linear functional. Finally, by using limit processes, we recover the classification of coherent pairs of linear functionals stated by Meijer. Ó

(1,1)-Dω-coherent pairs

“…This is a generalization of the results obtained by I. Area, et al, in [3,5] for (1, 0)-q-coherent pairs. They showed that if (U, V) is a (1, 0)-q-coherent pair of quasi-definite linear functionals then at least one of them must be q-classical and one is a rational modification of the other as above with deg(σ(x)) ≤ 2.…”

“…Conversely, if there are constants a n = 0 and c n (λ, ω), 1 ≤ n ≤ M , such that (3.3) holds, then there exist constants b n with 5) such that (3.1) holds, i.e., (U, V) is a (1, 1)-D ω -coherent pair.…”

On linearly related sequences of difference derivatives of discrete orthogonal polynomials

“…This is a generalization of the results obtained by I. Area, E. Godoy, and F. Marcellán ([2,4] for ν = ω, [3] for ν = q) for (1, 0)-D ν -coherent pairs. They proved that (1, 0)-D ν -coherence is a sufficient condition for at least one of the linear functionals to be D ν -classical and each of them to be a rational modification of the other as above with deg(σ(x)) ≤ 2.…”

“…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).…”

“…This appears to be an hard task from a technical point of view, and some examples are now under construction (which we hope to be the subject of further work) following ideas presented in previous works on coherent pairs of OPs, not only motivated by the continuous case, but also by the q−case. For instance, an important source of motivation is Section 6 contained in the paper [3] by I. 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.…”