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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