Orthogonal Polynomials and Special Functions

Francisco Marcellán is full professor of applied mathematics at the Universidad Carlos III de Madrid in Leganés (Spain) where he served as vice rector of research. In the past he has been teaching at the universities of Zaragoza, Santiago de Compostela and the Polytécnica de Madrid and was recently a visiting professor at the Georgia Institute of Technology. He was program director of the SIAM activity group on Orthogonal Polynomials and Special Functions from 1999 to 2005. Presently he is the director of the Spanish National Agency for Quality Assurance and Accreditation (ANECA). He has several publications in mathematical analysis (most particularly special functions and approximation) and linear algebra.

Walter Van Assche is full professor of mathematics at the Katholieke Universiteit Leuven in Belgium and was a research director of the Belgian National Fund for Scientific Research. He was vice chair of the SIAM activity group on Orthogonal Polynomials and Special Functions from 1999 to 2005 and is on the editorial board of the Journal of Approximation Theory and the Journal of Difference Equations and Applications. He is the author of Asymptotics for Orthogonal Polynomials (Lecture Notes in Mathematics 1265) and wrote two chapters in M.E.H. Ismail's book Classical and Quantum Orthogonal Polynomials in one Variable.

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.