a b s t r a c tA pair of regular linear functionals (U, V) is said to be a (M, N)-coherent pair of order (m, k) if their corresponding sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 satisfy a structure relation such as To be more precise, let us assume that m ≥ k. If m = k then U and V are related by a rational factor (in the distributional sense); if m > k then U and V are semiclassical and they are again related by a rational factor. In the second part of this work we deal with a Sobolev type inner product defined in the linear space of polynomials with real coefficients, P, aswhere λ is a positive real number, m is a positive integer number and (μ 0 , μ 1 ) is a (M, N)-coherent pair of order m of positive Borel measures supported on an infinite subset of the real line, meaning that the sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 with respect to μ 0 and μ 1 , respectively, satisfy a structure relation as above with k = 0, a i,n and b i,n being real numbers fulfilling the above mentioned conditions. We generalize several recent results known in the literature in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs (introduced in [A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products J. Approx. Theory 65 (2) (1991) 151-175]) and their extensions. In particular, we show how to compute the coefficients of the Fourier expansion of functions on an appropriate Sobolev space (defined by the above inner product) in terms of the sequence of Sobolev orthogonal polynomials {S n (x; λ)} n≥0 .

Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let D ν be the corresponding difference operator defined in the usual way either by(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 ν of higher orders m and k (resp.) are connected by a linear algebraic structure relation such aswhere M, N, m, k ∈ 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 ν−distributional sense). Moreover, when m = k then both U and V are D νsemiclassical functionals. This leads us to the concept of (M, N )-D ν -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 productassuming that U and V (which, eventually, may be represented by discrete measures sup-ported either on a uniform lattice if ν = ω, or on a q-lattice if ν = q) constitute a (M, N )-D ν -coherent pair of order m (that is, an (M, N )-D ν -coherent pair of order (m, 0)), m ∈ N being fixed.

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In this work, we introduce the notion of (1, 1)-Dω-coherent pair of weakly quasi-definite linear functionals (U, V) as the Dω-analogue to the generalized coherent pair studied by A. Delgado and F. Marcellán in [8]. This means that their corresponding families of monic orthogonal poly-We prove that (1, 1)-Dω-coherence is a sufficient condition for the weakly quasi-definite linear functionals to be Dω-semiclassical, one of them of class at most 1 and the another of class at most 5, and they are related by a expression of rational type. Additionally, a matrix interpretation of (1, 1)-Dω-coherence in terms of the corresponding monic Jacobi matrices is given. The particular case when U is Dω-classical linear functional is studied.

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