Walter Van Assche - Böcker

There are a number of intriguing connections between Painlevé equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlevé equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlevé transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlevé equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlevé equations.

Recently there has been a great deal of interest in the theory of orthogonal polynomials. The number of books treating the subject, however, is limited. This monograph brings together some results involving the asymptotic behaviour of orthogonal polynomials when the degree tends to infinity, assuming only a basic knowledge of real and complex analysis. An extensive treatment, starting with special knowledge of the orthogonality measure, is given for orthogonal polynomials on a compact set and on an unbounded set. Another possible approach is to start from properties of the coefficients in the three-term recurrence relation for orthogonal polynomials. This is done using the methods of (discrete) scattering theory. A new method, based on limit theorems in probability theory, to obtain asymptotic formulas for some polynomials is also given. Various consequences of all the results are described and applications are given ranging from random matrices and birth-death processes to discrete Schrödinger operators, illustrating the close interaction with different branches of applied mathematics.

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.

OrthogonalPolynomialsandSpecialFunctions(OPSF)is a veryoldbranchof mathematics having a very rich history. Many famous mathematicians have contributed to the subject: Euler's work on the gamma function, Gauss's and Riemann's work onthe hypergeometricfunctions andthe hypergeometric di?erentialequation,Abel's andJacobi'sworkonelliptic functions,andsoon. Usuallythespecialfunctionshavebeenintroducedto solveaspeci?cproblem, and many of them occurred in solving the di?erential equations describing a physical problem, e.g., the astronomerBessel introduced the functions named afterhiminhisworkonKepler'sproblemofthreebodiesmovingundermutual gravitation. So the subject OPSF is very classical and there have been very interesting developments through the centuries, and there have been numerous appli- tions to various branches of mathematics, e.g. combinatorics, representation theory,numbertheory,andapplicationstophysicsandastronomy,suchasthe afore-mentioned classical physical problems, but also integrable systems, - tics,quantumchemistry,etcetera.SoOPSFiswell-established,andverymuch driven by applications. The advent of the computer, ?rst thought to be fatal to the subject, turned out to be a stimulus, ?rst of all because it allowedmore detailed computations requiring special numerical algorithms, but mainly - cause it led to automatic summation routines, notably the WZ-method (see Koepf's contribution). So OPSF is a very lively branch of mathematics. Since more advanced courses on OPSF seldom appear in the curriculum, we felt the need for such courses for young researchers (graduate students and post-docs). A series of European summer schools was started with one in Laredo, Spain (2000) and one in Inzell, Germany (2001). This book contains the notes for the lectures of the summer school in Belgium in 2002, which took place from August 12-16, 2002, at the Katholieke Universiteit Leuven, Belgium.In2003asummerschoolinOPSFwillbeheldinCoimbra,Portugal.