Orthogonal Polynomials and Painlevé Equations

by Walter Van Assche

Overview

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.

Product Details

ISBN-13:	9781108608978
Publisher:	Cambridge University Press
Publication date:	12/28/2017
Series:	Australian Mathematical Society Lecture Series , #27
Sold by:	Barnes & Noble
Format:	eBook
File size:	8 MB

About the Author

Walter Van Assche is a professor of mathematics at the Katholieke Universiteit Leuven, Belgium, and presently the Chair of the SIAM Activity Group on Orthogonal Polynomials and Special Functions (OPSF). He is an expert in orthogonal polynomials, special functions, asymptotics, approximation, and recurrence relations.

Table of Contents

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.