Uniform asymptotics for orthogonal polynomials via the Riemann--Hilbert approach

Abstract

In this survey, a brief description is given to the Riemann--Hilbert approach for uniform asymptotics of orthogonal polynomials. The Jacobi polynomials are used to demonstrate the steps. Some essentially new progress of the approach is reported. A short discussion on prospective development is also included.

†Dedicated to Professor Wei Lin on the occasion of his 70th birthday.

Keywords:

• Orthogonal polynomials
• Uniform asymptotics
• Riemann--Hilbert approach

Keywords:

• 2000 Mathematics Subject Classifications: 41A60
• 33C45

Acknowledgements

The author is grateful to Prof. R. Wong and Dr. Z. Wang for helpful discussion. This research was partially supported by the Natural Science Foundation of China, GuangDong Natural Science Foundation and a fund from the Advanced Research Center of ZhongShan University.

Notes

†Dedicated to Professor Wei Lin on the occasion of his 70th birthday.