On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Let v be either omega is an element of C {0} or q is an element of C \ {0, 1}, and let D-v 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-v of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as