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19
Orthogonal polynomials on the unit circle, qGamma weights, and discrete Painlevé equations
, 2009
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On the Lagrangian Structure of the Discrete Isospectral and Isomonodromic Transformations
"... Abstract. We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the refactorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the MoserVeselov approach to ..."
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Abstract. We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the refactorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the MoserVeselov approach to integrability of discrete systems via the refactorization of matrix polynomials to a more general class of matrix rational functions that have a simple divisor and, in the quadratic case, explicitly write the Lagrangian function for such systems. Next we show that if we let certain parameters in this Lagrangian to be timedependent, the resulting EulerLagrange equations describe the isomonodromic transformations of systems of linear difference equations. It is known that in some special cases such equations reduce to the difference Painlevé equation. As an example, we show how to obtain the difference Painlevè V equation in this way, and hence we establish that this equation can be written in the Lagrangian form. 1.
Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table and analogs of isomonodromy
, 2012
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Symmetries and special solutions of reductions of the lattice potential KdV equation
"... Abstract. We identify a periodic reduction of the nonautonomous lattice potential Korteweg–de Vries equation with the additive discrete Painlevé equation with E (1) 6 symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the disc ..."
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Abstract. We identify a periodic reduction of the nonautonomous lattice potential Korteweg–de Vries equation with the additive discrete Painlevé equation with E (1) 6 symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation. Key words: difference equations; integrability; reduction; isomonodromy
An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations) ⋆
"... Abstract. We construct a family of secondorder linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higherorder analogues), and admitting a large family of monodromypreserving deformations. The solutions are certain semiclassical biorthogon ..."
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Abstract. We construct a family of secondorder linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higherorder analogues), and admitting a large family of monodromypreserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higherorder analogues of Spiridonov’s elliptic beta integral. Key words: isomonodromy; hypergeometric; Painlevé; biorthogonal functions 2010 Mathematics Subject Classification: 33E17; 34M55; 39A13 1
TAUFUNCTION OF DISCRETE ISOMONODROMY TRANSFORMATIONS AND PROBABILITY
, 2007
"... We introduce the τfunction of a rational dconnection and its isomonodromy transformations. We show that in a continuous limit our τfunction agrees with the JimboMiwaUeno τfunction, compute the τfunction for the isomonodromy transformations leading to difference Painlevé V and difference Painl ..."
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We introduce the τfunction of a rational dconnection and its isomonodromy transformations. We show that in a continuous limit our τfunction agrees with the JimboMiwaUeno τfunction, compute the τfunction for the isomonodromy transformations leading to difference Painlevé V and difference Painlevé VI equations, and prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the τfunction for an associated dconnection.
Discrete Hamiltonian Structure of Schlesinger Transformations
, 2013
"... Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transf ..."
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Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painleve ́ equations; this is also the main theme behind our work. In this paper we show how to write an elementary Schlesinger transformation as a discrete Hamiltonian system w.r.t. the standard symplectic structure on the space of Fuchsian systems. We then show how Schlesinger transformations reduce to discrete Painleve ́ equations by considering two explicit examples, dP D