Orthogonal Polynomials, Quadrature, and Approximation: Computational Methods and Software (in Matlab) -- Equilibrium Problems of Potential Theory in the Complex Plane -- Discrete Orthogonal Polynomials and Superlinear Convergence of Krylov Subspace Methods in Numerical Linear Algebra -- Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case -- Orthogonal Polynomials and Separation of Variables -- An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials -- Painlevé Equations — Nonlinear Special Functions.

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askey’s scheme using Leonard pairs, and non-linear special functions associated with the Painlevé equations.