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On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Opened Access On linearly related sequences of difference derivatives of discrete orthogonal polynomials

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Autor: Álvarez Nodarse, Renato
Soares Petronilho, José Carlos
Pinzón Cortés, Natalia Camila
Sevinik Adigüzel, Rezan
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2015-08-15
Publicado en: Journal of Computational and Applied Mathematics, 284, 26-37.
Tipo de documento: Artículo
Resumen: 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 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)-...
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Cita: 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.
Tamaño: 434.1Kb
Formato: PDF

URI: http://hdl.handle.net/11441/41765

DOI: http://dx.doi.org/10.1016/j.cam.2014.06.018

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