In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle. most 0, 1, and 2, respectively. He also found that the Hermite, Jacobi, and Laguerre families are the only polynomial solutions for (1) with a corresponding measure that is positive. Moreover, such sequences satisfy the so-called Hahn's property: their derivatives also constitute an orthogonal family. It is interesting that some of these families have been studied in the context of symmetry Lie algebras. For instance, in [3][4][5][6], the authors studied how the differential equations satisfied by the classical families can be obtained from the second-order Casimir elements of the corresponding symmetry algebra and of some of its subalgebras. Furthermore, some operators defined on Lie algebras are used in [7] to obtain differential properties of some special functions, including the Jacobi polynomials.On the other hand, in the last decades, some canonical examples of spectral transformations of orthogonality measures have been studied in the literature: the Christoffel transformation, consisting in a polynomial modification of the measure; the Uvarov transformation, defined by the addition of a Dirac's delta measure; and the Geronimus transformation, where the orthogonality measure is divided by a polynomial, and a Dirac's delta measure is added. For instance, in [8], the author is interested in the relations between the associated Stieltjes functions. Furthermore, he shows that all linear spectral transformations of Stieltjes functions can be obtained as a finite product of the three canonical transformations mentioned above. On the other hand, in [9], the authors consider the relations between these perturbations and LU factorizations of the corresponding Jacobi matrices, in the more general framework of orthogonality with respect to linear functionals. The study of differential equations of higher order satisfied by orthogonal polynomials was initiated by H. L. Krall in [10] and later on A. M. Krall (see [11]) showed that the sequences orthogonal with respect to an Uvarov perturbation of the classical families satisfy a fourth order differential equation. The orthogonal families associated with these perturbations are often called classical-type orthogonal polynomials. It is important to notice that they are not longer classical, and the polynomial coefficients in the corresponding differential equations (which are called holonomic equations) can even depend on the degree n (see [12]), in sharp contrast with the classical case...

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In this paper, we show how to compute in Oðn 2 Þ steps the Fourier coefficients associated with the Gelfand Levitan approach for discrete Sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located out side the closed unit disk. As a consequence, we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure. Moreover, we show how to recover the discrete part of our Sobolev inner product.

In this contribution, we use the connection between stable polynomials and orthogonal polynomials on the real line to construct sequences of Hurwitz polynomials that are robustly stable in terms of several uncertain parameters. These sequences are constructed by using properties of orthogonal polynomials, such as the well-known three-term recurrence relation, as well as by considering linear combinations of two orthogonal polynomials with consecutive degree. Some examples are presented.

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