On linearly related sequences of difference derivatives of discrete orthogonal polynomials

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 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 ∑i=0Mai,nDνmPn+m−i(x)=∑i=0Nbi,nDνkQn+k−i(x),n≥0, 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,(Dνmp)(x)(Dνmr)(x)〉,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.