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.

Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem / Bertola, M.; Eynard, B.; Harnad, J.. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 243:2(2003), pp. 193-240. [10.1007/s00220-003-0934-1]

Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem

Bertola, M.;
2003

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.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
243
2
193
240
https://link.springer.com/article/10.1007%2Fs00220-003-0934-1
https://arxiv.org/abs/nlin/0208002
Bertola, M.; Eynard, B.; Harnad, J.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/14975
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