Article
On linearly related sequences of difference derivatives of discrete orthogonal polynomials
Author/s | Álvarez Nodarse, Renato
Soares Petronilho, José Carlos Pinzón Cortés, Natalia Camila Sevinik Adigüzel, Rezan |
Department | Universidad de Sevilla. Departamento de Análisis Matemático |
Publication Date | 2015-08-15 |
Deposit Date | 2016-06-01 |
Published in |
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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 ... 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)-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. |
Project ID. | info:eu-repo/grantAgreement/MINECO/MTM2012-36732-C03
FQM-262 FQM-7276 P09-FQM-4643 PEst-C/MAT/UI0324/2013 |
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. |
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