On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Let νν be either ω∈C∖{0}ω∈C∖{0} or q∈C∖{0,1}q∈C∖{0,1}, and let DνDν 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 UU and VV be two moment regular linear functionals and let {Pn(x)}n≥0{Pn(x)}n≥0 and {Qn(x)}n≥0{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{Pn(x)}n≥0 and {Qn(x)}n≥0{Qn(x)}n≥0 assuming that their difference derivatives DνDν of higher orders mm and kk (resp.) are connected by a linear algebraic structure relation such as ∑i=0Mai,nDνmPn+m−i(x)=∑i=0Nbi,nDνkQn+k−i(x),n≥0, where M,N,m,k∈N∪{0}M,N,m,k∈N∪{0}, aM,n≠0aM,n≠0 for n≥Mn≥M, bN,n≠0bN,n≠0 for n≥Nn≥N, and ai,n=bi,n=0ai,n=bi,n=0 for i>ni>n. Under certain conditions, we prove that UU and VV are related by a rational factor (in the ν−ν−distributional sense). Moreover, when m≠km≠k then both UU and VV are DνDν-semiclassical functionals. This leads us to the concept of (M,N)(M,N)-DνDν-coherent pair of order (m,k)(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,(Dνmp)(x)(Dνmr)(x)〉,λ>0, assuming that UU and VV (which, eventually, may be represented by discrete measures supported either on a uniform lattice if ν=ων=ω, or on a qq-lattice if ν=qν=q) constitute a (M,N)(M,N)-DνDν-coherent pair of order mm (that is, an (M,N)(M,N)-DνDν-coherent pair of order (m,0)(m,0)), m∈Nm∈N being fixed.