Abstract
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.
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Communicated by L. Takhtajan
Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Québec
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Bertola, M., Eynard, B. & Harnad, J. Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem. Commun. Math. Phys. 243, 193–240 (2003). https://doi.org/10.1007/s00220-003-0934-1
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DOI: https://doi.org/10.1007/s00220-003-0934-1