Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem

We consider biorthogonal polynomials that arise in the study of a generalization of two-matrix Hermitian models with two polynomial potentials V-1(x), V-2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials ("windows"), of lengths equal to the degrees of the potentials V-1 and V-2, satisfy systems of ODE's with polynomial coefficients as well as PDE's (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.