Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials

A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integer. From this orthogonality, we derive several equivalent Riemann–Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique.