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There are a number of intriguing connections between Painleve 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 Painleve equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painleve transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painleve 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 Painleve equations.

Orthogonal Polynomials and Painleve Equations

1. Introduction; 2. Freud weights and discrete Painleve I; 3. Discrete Painleve II; 4. Ladder operators; 5. Other semi-classical orthogonal polynomials; 6. Special solutions of Painleve equations; 7. Asymptotic behavior of orthogonal polynomials near critical points; Appendix. Solutions to exercises; References; Index.