Asymptotics of Orthogonal Polynomials and Painleve Transcendents

Description

Orthogonal polynomials play an important role in many branches of mathematics, including mathematical physics, approximation theory and combinatorics. One of the major topics in the study of this area is to find the asymptotic behavior of these orthogonal polynomials as their degree tends to infinity. There have been a lot of asymptotic results for a quite large class of orthogonal polynomials in the literature. Their asymptotic expansions usually involve trigonometric functions and special functions like Airy functions, Bessel functions and parabolic cylinder functions. Recently, it was found that in critical situations a new class of functions, Painlevétranscendents, appears in the local asymptotics of orthogonal polynomials near singular points. Although breakthroughs have been made in the recent years, there are still many important open problems in this area.In this project, we will focus on problems related to the asymptotics of orthogonal polynomials and further investigate their relations with Painlevétranscendents.