On Linearly Related Sequences of Difference Derivatives of Discrete Orthogonal Polynomials.

Let \nu be either \omega\in C\{0} or q\in C\{0,1}, and let D_\nu be the corresponding difference operator defined in the usual way either by D_\omega p(x) = [p(x+\omega)-p(x) / [\omega or D_q p(x) = [p(qx)-p(x) / [(q-1)x. Let U and V be two moment regular linear functionals and let {P_n(x)}_{n>=0} and {Q_n(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 {P_n(x)}_{n>=0} and {Q_n(x)}_{n>=0} assuming that their difference derivatives D_\nu of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as \sum_{i=0}^M a_{i,n} D_\nu^m P_{n+m-i}(x) = \sum_{i=0}^N b_{i,n} D_\nu^k Q_{n+k-i}(x), n>=0, where M, N, m, k \in N\{0}, a_{M,n}>=0 for n>= M, b_{N,n}>=0 for n>=N, and a_{i,n}=b_{i,n}=0 for i>n. Under certain conditions, we prove that U and V are related by a rational factor (in the \nu-distributional sense). Moreover, when m =! k then both U and V are D_\nu-semiclassical functionals. This leads us to the concept of (M,N)-D_\nu-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 _{\lambda,\nu} = + \lambda , \lambda>0, assuming that U and V (which, eventually, may be represented by discrete measures supported either on a uniform lattice if \nu=\omega, or on a q-lattice if \nu=q) constitute a (M,N)-D_\nu-coherent pair of order m (that is, an (M,N)-D_\nu-coherent pair of order (m,0)), m\in N being fixed.