# V Congreso de la Red de Polinomios Ortogonales y Teoría de Aproximación - Orthonet

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Publication Open Access Shannon-like integrals of hypergeometric orthogonal polynomials with large parameters and applications to high-dimensional harmonic and hydrogenic systems(2019) Valero Toranzo, IreneShow more In this talk we determine the asymptotics of various logarithmic-type integral functionals of hypergeometric orthogonal polynomials (Laguerre, Gegenbauer) when their parameter α -> ∞. Then, we apply the corresponding results to find the physical Shannon entropies for all the stationary states of harmonic and hydrogenic systems with a very high dimensionality D. Briefly, it is found that these entropies have the same rate of growth, O (D log D), when D -> ∞ 1 for both types of quantum systems.Show more Publication Open Access Coherent pairs of bivariate orthogonal polynomials(2019) Marcellán, Francisco; Marriaga, Misael E.; Pérez, T. E.; Piñar, M. A.Show more Coherent pairs of measures were introduced in 1991 and constitute a very useful tool in the study of Sobolev orthogonal polynomials on the real line. In this work, coherence and partial coherence in two variables appear as the natural extension of the univariate case. Given two families of bivariate orthogonal polynomials expressed as polynomial systems, they are a partial coherent pair if there exists a polynomials of the second family can be given as a linear combination of the _x001C_rst partial derivatives of (at most) three consecutive polynomials of the _x001C_rst family. A full coherent pair is a pair of families of bivariate orthogonal polynomials related by means of partial coherent relations in each variable. Consequences of this kind of relations concerning both families of bivariate orthogonal polynomials are studied.Show more Publication Open Access Computational methods for cumulative distribution functions(2019) Gil, AmparoShow more Some special functions are particularly relevant in Applied Probability and Statistics. For example, the incomplete gamma and beta functions are (up to normalization factors) the cumulative central gamma and beta distribution functions, respectively. The corresponding noncentral distributions (the Marcum-Q function and the cumulative noncentral beta distribution function) play also a signi_x001C_cant role in several applications. The inversion of cumulative distribution functions (CDFs) is also an important problem, in particular for computing percentage points or values of some relevant parameters when the distribution function is involved in hypothesis testing. In this talk, methods for computing and inverting the gamma and beta CDFs are discussed. The performance of the methods will be illustrated with numerical examples. As we will see, we may contemplate CDFs as a branch of the large family of special functions yet probably not so well known as other classical functions.Show more Publication Open Access Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis(2019) Temme, Nico M.Show more In this talk we describe a simpler quadrature method, in fact, the trapezoidal rule, which appears to be very efficient on the saddle point contour, and on a slightly shifted one.Show more Publication Open Access Discrete harmonic analysis associated with Jacobi expansions(2019) Arenas, Alberto; Ciaurri, Óscar; Labarga, EdgarShow more The study of the classical harmonic analysis operators in non-trigonometric contexts has a very rich history and it has been widely addressed in the continuous setting. However, the situation in the discrete one is totally opposed. In spite of several works in the discrete setting (v.g. [5]), there was a lack of them in non-trigonometric contexts until the paper by J. J. Betancor et al. [4], where the ultraspherical orthonormal system is considered. In this talk, we extend that work and we present the study of classical Harmonic Analysis operators associated with Jacobi expansions.Show more Publication Open Access Orthogonality and bispectrality(2019) Durán Guardeño, AntonioShow more The concept of bispectrality (in short, a function in two variables that is an eigenfunction for an operator in each variable) is especially interesting for orthogonal polynomials. Indeed, depending on the type of operators (differential, difference, q-difference, etc.) and their orders, the bispectrality characterizes the most important families of orthogonal polynomials, from the classical, classical discrete or q -classical polynomials, to the Krall and exceptional polynomials. In my opinion, one of the most interesting (and difficult) problems in relation to orthogonality and bispectrality is the characterization of the two algebras associated with each family of bispectral polynomials. In this talk I will review the state of the art about this problem.Show more Publication Open Access Sobolev inner product as a solution of inverse Darboux transformation(2019) Cantero, María JoséShow more In this talk we expose a survey of Darboux transformations for Jacobi and CMV matrices, relating them with orthogonal polynomials on the real line, orthogonal polynomials on the unit circle and integrable systems, and also highlighting their similarities and di_x001B_erences. We will show the interest of the above mentioned spurious solutions of inverse Darboux for CMV matrices: they are related to Sobolev type inner products.Show more