Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-05-03
J. Approx. Theory 144 (2007) no. 2, 162-212
Nonlinear Sciences
Exactly Solvable and Integrable Systems
47 pages, 4 figures
Scientific paper
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 on an 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 on 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 be interpreted as a pairing between two dual Riemann-Hilbert problems.
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