On linearly related sequences of difference derivatives of discrete orthogonal polynomials

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dc.contributor.authorAlvarez-Nodarse, R.
dc.contributor.authorPetronilho, J.
dc.contributor.authorPinzon-Cortes, N. C.
dc.contributor.authorSevinik-Adiguzel, R.
dc.date.accessioned2020-03-26T19:06:30Z
dc.date.available2020-03-26T19:06:30Z
dc.date.issued2015
dc.departmentSelçuk Üniversitesien_US
dc.description.abstractLet 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.en_US
dc.description.sponsorshipDireccion General de Investigacion, Desarrollo e Innovacion; Ministerio de Economia y Competitividad of Spain [MTM2012-36732-C03]; Junta de Andalucia (Spain)Junta de Andalucia [FQM262, FQM-7276, P09-FQM-4643]; FEDER fundsEuropean Union (EU); Centro de Matematica da Universidade de Coimbra (CMUC) - European Regional Development Fund through the program COMPETE; Portuguese Government through the FCT Fundacao para a Ciencia e a Tecnologia [PEst-C/MAT/UI0324/2013]; TUBITAK, the Scientific and Technological Research Council of TurkeyTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK)en_US
dc.description.sponsorshipWe are grateful to Prof. Francisco Marcellan for his valuable comments and remarks that helped us to improve the paper. This work was supported by Direccion General de Investigacion, Desarrollo e Innovacion, Ministerio de Economia y Competitividad of Spain, under grants MTM2012-36732-C03 (RAN, NCP-C, JP), Junta de Andalucia (Spain) under grants FQM262, FQM-7276, and P09-FQM-4643 (RAN), FEDER funds (RAN). The work of J. Petronilho was also partially supported by the Centro de Matematica da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT Fundacao para a Ciencia e a Tecnologia under the project PEst-C/MAT/UI0324/2013. The work of R. Sevinik was supported by TUBITAK, the Scientific and Technological Research Council of Turkey.en_US
dc.identifier.doi10.1016/j.cam.2014.06.018en_US
dc.identifier.endpage37en_US
dc.identifier.issn0377-0427en_US
dc.identifier.issn1879-1778en_US
dc.identifier.scopusqualityQ2en_US
dc.identifier.startpage26en_US
dc.identifier.urihttps://dx.doi.org/10.1016/j.cam.2014.06.018
dc.identifier.urihttps://hdl.handle.net/20.500.12395/32384
dc.identifier.volume284en_US
dc.identifier.wosWOS:000351792100003en_US
dc.identifier.wosqualityQ1en_US
dc.indekslendigikaynakWeb of Scienceen_US
dc.indekslendigikaynakScopusen_US
dc.language.isoenen_US
dc.publisherELSEVIER SCIENCE BVen_US
dc.relation.ispartofJOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICSen_US
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US
dc.rightsinfo:eu-repo/semantics/openAccessen_US
dc.selcuk20240510_oaigen_US
dc.subjectOrthogonal polynomialsen_US
dc.subjectInverse problemsen_US
dc.subjectSemiclassical orthogonal polynomialsen_US
dc.subjectCoherent pairsen_US
dc.subjectSobolev-type orthogonal polynomialsen_US
dc.titleOn linearly related sequences of difference derivatives of discrete orthogonal polynomialsen_US
dc.typeArticleen_US

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