The PDEs of biorthogonal polynomials arising in the two-matrix model

Details The PDEs of biorthogonal polynomials arising in the two-matrix model The PDEs of biorthogonal polynomials arising in the two-matrix model

2003-11-18

arxiv.org/abs/nlin/0311033v2

Math. Phys. Anal. Geom. 9 (2006), no. 1, 23-52

Nonlinear Sciences

Exactly Solvable and Integrable Systems

20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake corrected

Scientific paper

The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (Delta-E) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of bi-orthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.

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