Citation:
Journal of Computational and Applied Mathematics, 2015, v. 284, pp. 26–37

ISSN:
0377-0427

DOI:
10.1016/j.cam.2014.06.018

Sponsor:
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)

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≥0Let 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,
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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,
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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.[+][-]

Description:

Proceedings of: OrthoQuad 2014. Puerto de la Cruz, Tenerife, Spain. January 20–24, 2014