RT Conference Proceedings
T1 On linearly related sequences of difference derivatives of discrete orthogonal polynomials
A1 Álvarez-Nodarse, Renato
A1 Petronilho, José
A1 Pinzón-Cortés, Natalia Camila
A1 Sevinik-Adıgüzel, Rezan
AB 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 offwhere  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 offassuming 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.
PB Elsevier
SN 0377-0427
YR 2015
FD 2015-08-15
LK https://hdl.handle.net/10016/23405
UL https://hdl.handle.net/10016/23405
LA eng
NO Proceedings of: OrthoQuad 2014. Puerto de la Cruz, Tenerife, Spain. January 20–24, 2014
NO 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 Competitividadof 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)
DS e-Archivo
RD 17 jul. 2024