Biorthogonal Systems on the Unit Circle, Regular Semiclassical Weights, and the Discrete Garnier Equations
نویسندگان
چکیده
منابع مشابه
Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations
We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrenc...
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We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z 1 ,. .. ...
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Abstract. The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has a...
متن کاملCommutation Relations and Discrete Garnier Systems
We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the fact...
متن کاملMatrix biorthogonal polynomials on the unit circle and non-abelian Ablowitz-Ladik hierarchy
In [13] Adler and van Moerbeke described a reduction of 2D-Toda hierarchy called Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik [1] using semidiscrete zero-curvature equations. In this paper we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computatio...
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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2008
ISSN: 1687-0247,1073-7928
DOI: 10.1093/imrn/rnn152