Special Functions and Orthogonal Polynomials, 2nd Edition

AUTHORS:
Richard Beals, Yale University, Connecticut
Roderick Wong, City University of Hong Kong



The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.

• Covers standard topics from a unified point of view to show how different topics are part of a general scheme
• Comprehensive but self-contained, covering newer asymptotic methods to give an up-to-date view of an important research area
• Includes topics such as Painlevé functions and Meijer G-functions, which are not usually treated at this level, to give an understandable and well-motivated introduction to some subjects of great current interest

Table of Contents

1. Orientation
2. Gamma, beta, zeta
3. Second-order differential equations
4. Orthogonal polynomials on an interval
5. The classical orthogonal polynomials
6. Semiclassical orthogonal polynomials
7. Asymptotics of orthogonal polynomials: two methods
8. Confluent hypergeometric functions
9. Cylinder functions
10. Hypergeometric functions
11. Spherical functions
12. Generalized hypergeometric functions
G-functions
13. Asymptotics
14. Elliptic functions
15. Painlevé transcendents
Appendix A. Complex analysis
Appendix B. Fourier analysis
References
Index.