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arXiv · 0811.4027

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

N. S. Witte
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Abstract

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, ..., z_M \} $ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $ τ$-functions or equivalently Toeplitz determinants of the system.

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N. S. Witte. 2008-11-25. Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II. https://arxiv.org/abs/0811.4027

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