Biorthogonal polynomials for two-matrix models with semiclassical potentials

摘要

We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V-1, V-2 with arbitrary rational derivative and whose supports are constrained oil all arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms d(i) depending on the number of hard-edges and oil the degree of the rational functions V-i'. Using these relations we derive Christoffel-Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulae for the differential equation satisfied by d(i) + 1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann-Hilbert problem for (d(i) + 1) x (d(i) + 1) matrices constructed out of the polynomials and these transforms. Moreover, we prove that the Christoffel-Darboux pairing can ne interpreted is a pairing between two dual Riemann-Hilbert problems. (C) 2006 Elsevier Inc. All rights reserved.