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 ( p n ) n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k ≥ 2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (q n ) n and two polynomials π k and θ m , with degrees k and m (resp.), with 0 ≤ m ≤ k − 1, such that p nk+m (x) = θ m (x)q n (π k (x)) (n = 0, 1, 2, . . .).In this work we establish algebraic conditions for the existence of a polynomial mapping in the above sense. Under such conditions, when ( p n ) n is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS ( p n ) n in terms of the orthogonality measure for the OPS (q n ) n . Some applications and examples are presented, recovering several known results in a unified way.

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P n ) n and (Q n ) n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such asfor all n = 0, 1, 2, . . . , where M and N are fixed nonnegative integer numbers, and r i,n and s i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P n ) n and (Q n ) n (resp.). Assuming 0 m k, we prove the existence of four polynomials Φ M+m+i and Ψ N+k+i , of degrees M + m + i and N + k + i (resp.), such thatthe (k − m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k = m, then u and v are connected by a rational modification. If k = m + 1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k > m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k − m with polynomial coefficients.

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