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

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 dc.contributor.author Álvarez-Nodarse, Renato dc.contributor.author Petronilho, José dc.contributor.author Pinzón-Cortés, Natalia Camila dc.contributor.author Sevinik-Adıgüzel, Rezan dc.date.accessioned 2016-07-21T08:42:40Z dc.date.available 2017-08-15T22:00:07Z dc.date.issued 2015-08-15 dc.identifier.bibliographicCitation Journal of Computational and Applied Mathematics, 2015, v. 284, pp. 26–37 dc.identifier.issn 0377-0427 dc.identifier.uri http://hdl.handle.net/10016/23405 dc.description Proceedings of: OrthoQuad 2014. Puerto de la Cruz, Tenerife, Spain. January 20–24, 2014 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)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 ∑Mi=0ai,nDmνPn+m−i(x)=∑Ni=0bi,nDkνQn+k−i(x),n≥0, Turn MathJax off where M,N,m,k∈N∪{0} , aM,n≠0 for n≥M , bN,n≠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≠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 ⟨p(x),r(x)⟩λ,ν=⟨U,p(x)r(x)⟩+λ⟨V,(Dmνp)(x)(Dmνr)(x)⟩,λ>0, Turn MathJax off 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 We are grateful to Prof. Francisco Marcellán for his valuable comments and remarks that helped us to improve the paper. This work was supported by Dirección General de Investigación, Desarrollo e Innovación, Ministerio de Economía y Competitividad of Spain, under grants MTM2012-36732-C03 (RAN, NCP-C, JP), Junta de Andalucía (Spain) under grants FQM262, FQM-7276, and P09-FQM-4643 (RAN), FEDER funds (RAN) dc.format.extent 12 dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Elsevier dc.rights © Elsevier 2015 dc.rights Atribución-NoComercial-SinDerivadas 3.0 España dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/ dc.subject.other Orthogonal polynomials dc.subject.other Inverse problems dc.subject.other Semiclassical orthogonal polynomials dc.subject.other Coherent pairs dc.subject.other Sobolev-type orthogonal polynomials dc.title On linearly related sequences of difference derivatives of discrete orthogonal polynomials dc.type article dc.type conferenceObject dc.relation.publisherversion http://dx.doi.org/10.1016/j.cam.2014.06.018 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1016/j.cam.2014.06.018 dc.rights.accessRights openAccess dc.relation.projectID Gobierno de España. MTM-2012-36732-C03-01 dc.type.version acceptedVersion dc.relation.eventdate January 20–24, 2014 dc.relation.eventplace Puerto de la Cruz, Tenerife, Spain dc.relation.eventtitle OrthoQuad 2014 dc.relation.eventtype proceeding dc.identifier.publicationfirstpage 26 dc.identifier.publicationlastpage 37 dc.identifier.publicationtitle Journal of Computational and Applied Mathematics dc.identifier.publicationvolume 284
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