On linearly related sequences of difference derivatives of discrete orthogonal polynomials
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Date
2015
Journal Title
Journal ISSN
Volume Title
Publisher
ELSEVIER SCIENCE BV
Access Rights
info:eu-repo/semantics/openAccess
Abstract
Let v be either omega is an element of C {0} or q is an element of C \ {0, 1}, and let D-v be the corresponding difference operator defined in the usual way either by D(omega)p(x) =p(x+omega)-p(x)/omega or D(q)p(x) = p(qx)-p(x)/(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-v of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as Sigma(M)(i=0) a(i,n)D(v)(m)P(n+m-i)(x) = Sigma(N)(i=0) b(i,n)D(v)(k)Q(n+k-i)(x), n >= 0, where M, N, m, k is an element of N boolean OR {0}, a(M,n) not equal 0 for n >= M, b(N,n) not equal 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 v-distributional sense). Moreover, when m not equal k then both u and v are D-v-semiclassical functionals. This leads us to the concept of (M, N)-D-v-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 product < p(x), r(x)>(lambda,v) = < 11, p(x)r (x)> + lambda < v, (D(v)(m)p)(x)(D(v)(m)r)(x)>, lambda > 0, assuming that u and v (which, eventually, may be represented by discrete measures supported either on a uniform lattice if v = omega, or on a q-lattice if v = q) constitute a (M, N)-D-v-coherent pair of order m (that is, an (M, N)-D-v-coherent pair of order (m, 0)), m is an element of N being fixed. (C) 2014 Elsevier B.V. All rights reserved.
Description
Keywords
Orthogonal polynomials, Inverse problems, Semiclassical orthogonal polynomials, Coherent pairs, Sobolev-type orthogonal polynomials
Journal or Series
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
WoS Q Value
Q1
Scopus Q Value
Q2
Volume
284
