Biorthogonal polynomials for two-matrix models with semiclassical potentials

We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V1,V2 with arbitrary rational derivative and whose supports are constrained on an arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms di depending on the number of hard-edges and on the degree of the rational functions . Using these relations we derive Christoffel–Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulæ for the differential equation satisfied by di+1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann–Hilbert problem for (di+1)×(di+1) matrices constructed out of the polynomials and these transforms. Moreover, we prove that the Christoffel–Darboux pairing can be interpreted as a pairing between two dual Riemann–Hilbert problems.