Representation of Christoffel–Darboux formula for random matrix with external source and constrained eigenvalues

In this paper, we obtain a new representation of Christoffel–Darboux formula for random matrix with external source and constrained eigenvalues. In order to simplify the Christoffel–Darboux formula, we define two types of orthogonal polynomials, and find out the close relation between Riemann–Hilbert problem and orthogonal polynomials. Furthermore, the Christoffel–Darboux formula can be expressed as a solution of the Riemann–Hilbert problem.