Orthogonal polynomials and special functions : computation and applications ; [lecture notes of the Fifth European Summer School on Orthogonal Polynomials and Special Functions, which was held at the Universidad Carlos III de Madrid, Leganés, Spain from

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. Front Matter....Pages i-xiv Orthogonal Polynomials, Quadrature, and Approximation: Computational Methods and Software (in Matlab)....Pages 1-77 Equilibrium Problems of Potential Theory in the Complex Plane....Pages 79-117 Discrete Orthogonal Polynomials and Superlinear Convergence of Krylov Subspace Methods in Numerical Linear Algebra....Pages 119-185 Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case....Pages 187-228 Orthogonal Polynomials and Separation of Variables....Pages 229-254 An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials....Pages 255-330 Painlevé Equations — Nonlinear Special Functions....Pages 331-411 Back Matter....Pages 413-422

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? The present set of lecture notes contains seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions.