Orthogonal Polynomials and Painlevé Equations

A leading authority on orthogonal polynomials details their relationships with Painlevé equations with clear proofs, examples, and exercises.

Walter Van Assche (Author)

9781108441940, Cambridge University Press

Paperback / softback, published 28 December 2017

190 pages, 25 b/w illus.
22.8 x 15.2 x 1.2 cm, 0.29 kg

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.

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

Subject Areas: Differential calculus & equations [PBKJ], Functional analysis & transforms [PBKF], Number systems [PBCN]