Abstract:

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear
in the theory of Riemann -Hilbert approach to asymptotic analysis
for orthogonal polynomials on a real interval introduced by Fokas,
Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to
asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.

Key words: Semi-conjugate, matrix boundary value problem, orthogonal polynomials, smooth Jordan curve