We prove that the Nevalinna-Pick algorithm provides different homeomorphisms between certain topological spaces of measures, analytic functions and sequences of complex numbers. This algorithm also yields a continued fraction expansion of every Schur function, whose approximants are identified. The approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants are studied. The above results permit us to obtain a Khrushchev's formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is also presented.
Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I
经典舒尔分析与圆上的正交多项式理论密切相关[Simon，2005]。我们在这里调查多点舒尔分析和正交有理函数之间的联系。具体而言，我们研究了收敛的壁有理函数通过发展一个合理的模拟Szeg \H o理论，在这种情况下，插值点可能会积累在单位圆。这导致我们推广[Khrushchev，2001]，[Bultheel et al.，1999]，并产生一种新的类型的渐近。
A generalization of Schur functions: applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators . We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces . Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schu…
A determinant identity for moments of orthogonal polynomials that implies Uvarov's formula for the orthogonal polynomials of rationally related densities
Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
这些是2018年10月5日至12日在杜阿拉（喀麦隆）举行的\textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications}教授的特殊多项式课程的讲义。他们总结的基本结果和建设的特殊多项式，在过去的十年中发展。此外，还给出了潘勒韦方程PIV及其高阶推广的有理解的构造的一些新结果，这些高阶推广属于$A_{2n}^{(1)}$ -Painlevé族.该构造基于薛定谔算子的缀饰链，其势是谐振子的有理扩展。这里介绍的一些材料（Sturm-Liouville算子，经典正交多项式，Darboux-Crum变换等）。是经典的，可以在许多教科书中找到，而一些结果（属，交错和循环玛雅图）是新的，并首次在这套讲义。
Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics
Instituto de Matemáticas and Departamento de Matematica Aplicada, Universidad de Granada（格拉纳达大学数学研究所与应用数学系）；School of Mathematics, Georgia Institute of Technology（佐治亚理工学院数学学院）
