In this talk we consider matrix orthogonal polynomials (MVOPs) on a finite interval of the real line, with Jacobi-type weights. We are particularly interested in the asymptotic behavior as the degree of the polynomials tends to infinity, that we study with the method of steepest descent applied to the corresponding Riemann-Hilbert problem, as described by Grünbaum, de la Iglesia and Martínez-Finkelshtein. We include examples motivated by group theory.

This is joint work with Arno Kuijlaars (KU Leuven, Belgium) and Pablo Román (Universidad de Córdoba, Argentina).

In this talk I will discuss on two new sets of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Confluent and Gauss' hypergeometric functions. The former on unbounded support and the latter with bounded support on the real line. It was recently shown in [3] that the latter are indeed random walk polynomials. In both cases, this type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in enumerative combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property.

References:
[1] H. Lima and A. Loureiro, Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, to appear in Stud. Appl. Math. arXiv:2001.06820.
[2] H. Lima and A. Loureiro, Multiple orthogonal polynomials associated with confluent hypergeometric functions, J. Approx. Theory, 260 (2020), pp 105484.
[3] A. Branquinho, J. E. Fernández-Díaz, A. Foulquié-Moreno and M. Mañas, Hypergeometric Multiple Orthogonal Polynomials and Random Walks, (2021) arXiv:2107.00770v2

In this contribution, we present some algebraic and analytic properties related to spectral transformations of orthogonality matrix measures on the unit circle, that constitute generalizations from well-known results on the scalar case. We focus our attention on the so-called Christoffel, Geronimus and Uvarov transformations, and deal with connection formulas, factorizations of block Hessenberg and CVM matrices, and relative asymptotics for the associated orthogonal matrix polynomials. This is a joint work with Edinson Fuentes.

Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac-Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac-Sylvester matrix. (Based on work in collaboration with Tamás Görbe.)

The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice.

Fractional Calculus is one of the areas where Special Functions naturally arise. We will characterize fractional Sobolev spaces of potential type on graded groups. The approach is based on the Littlewood-Paley g-function. This is a joint work with Pablo de Nápoli.

We present the Sobolev Type orthogonal polynomials of several variables defined on the cone and on the surface of the cone as a particular case of the polynomials on a quadratic surface of revolution. Using the results from [1], [2] and [3], we show that this kind of polynomials satisfy some properties.

[1] L. Fernández, T. E Pérez, M. Piñar and Y. Xu. Krall-type orthogonal polynomials in several variables. Comp. Appl. Math. Vol 81. Pags 1519-1524. (2001).
[2] Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory. 152 (2008), 52--65. (2006).
[3] Y. Xu, Fourier series in orthogonal polynomials on a cone of revolution, arXiv:1905.07587 2019

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegö polynomials. These results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions.

This is a joint work with Markus Hunziker, Taylor Poe, and Brian Simanek, all at Baylor University.