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# On linearly related sequences of difference derivatives of discrete orthogonal polynomials

 dc.creator Álvarez Nodarse, Renato es dc.creator Soares Petronilho, José Carlos es dc.creator Pinzón Cortés, Natalia Camila es dc.creator Sevinik Adigüzel, Rezan es dc.date.accessioned 2016-06-01T10:07:43Z dc.date.available 2016-06-01T10:07:43Z dc.date.issued 2015-08-15 dc.identifier.citation Álvarez Nodarse, R., Soares Petronilho, J.C., Pinzón Cortés, N.C. y Sevinik Adigüzel, R. (2015). On linearly related sequences of difference derivatives of discrete orthogonal polynomials. Journal of Computational and Applied Mathematics, 284, 26-37. dc.identifier.issn 0377-0427 es dc.identifier.uri http://hdl.handle.net/11441/41765 dc.description.abstract Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x) = p(x+ω)−p(x) es ω or Dqp(x) = p(qx)−p(x) (q−1)x. Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(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 {Pn(x)}n≥0 and {Qn(x)}n≥0 assuming that their difference derivatives Dν of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as X M i=0 ai,nDm ν Pn+m−i(x) = X N i=0 bi,nDk νQn+k−i(x), n ≥ 0, where M, N, m, k ∈ N ∪ {0}, aM,n 6= 0 for n ≥ M, bN,n 6= 0 for n ≥ N, and ai,n = bi,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 6= k then both U and V are Dν-semiclassical functionals. This leads us to the concept of (M, N)-Dν-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 hp(x), r(x)iλ,ν = hU, p(x)r(x)i + λ hV,(Dm ν p)(x)(Dm ν r)(x)i, λ > 0, assuming that U and V (which, eventually, may be represented by discrete measures supported either on a uniform lattice if ν = ω, or on a q-lattice if ν = q) constitute a (M, N)-Dν-coherent pair of order m (that is, an (M, N)-Dν-coherent pair of order (m, 0)), m ∈ N being fixed. dc.description.sponsorship Ministerio de Economía y Competitividad es dc.description.sponsorship Junta de Andalucía es dc.description.sponsorship Fondo Europeo de Desarrollo Regional es dc.description.sponsorship Fundaçao para a Ciência e a Tecnologia (Portugal) es dc.description.sponsorship Scientific and Technological Research Council of Turkey es dc.format application/pdf es dc.language.iso eng es dc.publisher Elsevier es dc.relation.ispartof Journal of Computational and Applied Mathematics, 284, 26-37. dc.rights Attribution-NonCommercial-NoDerivatives 4.0 Internacional * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject Orthogonal polynomials es dc.subject inverse problems es dc.subject semiclassical orthogonal polynomials es dc.subject coherent pairs es dc.subject Sobolev-type orthogonal polynomials es dc.title On linearly related sequences of difference derivatives of discrete orthogonal polynomials es dc.type info:eu-repo/semantics/article es dc.type.version info:eu-repo/semantics/submittedVersion es dc.rights.accessrights info:eu-repo/semantics/openAccess es dc.contributor.affiliation Universidad de Sevilla. Departamento de Análisis Matemático es dc.relation.projectID info:eu-repo/grantAgreement/MINECO/MTM2012-36732-C03 es dc.relation.projectID FQM-262 es dc.relation.projectID FQM-7276 es dc.relation.projectID P09-FQM-4643 es dc.relation.projectID PEst-C/MAT/UI0324/2013 es dc.relation.publisherversion http://dx.doi.org/10.1016/j.cam.2014.06.018 dc.identifier.doi 10.1016/j.cam.2014.06.018 es idus.format.extent 20 p. es dc.journaltitle Journal of Computational and Applied Mathematics es dc.publication.volumen 284 es dc.publication.initialPage 26 es dc.publication.endPage 37 es dc.identifier.idus https://idus.us.es/xmlui/handle/11441/41765
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