Alexander V. Kitaev

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Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of iso-monodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Bely˘ ı functions. Moreover, deformations(More)
We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. This deenition provides a uniied approach to the theories of \linear" special functions, i.e., classical higher transcen-dental functions, and \non-linear"(More)
We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to auxiliary ODEs with respect to a spectral parameter which associated with each SFIT. We consider two applications of(More)
We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of RS-transformations. Short title: Classification of RS 2 4 (3)-Transformations
We consider a linear 2 × 2 matrix ODE with two coalescing regular singulari-ties. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in(More)
Canonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik-Zamolodchikov equations. One can solve these equations by the OO-shell Bethe Ansatz method provided the Knizhnik-Zamolodchikov equations are related with the highest weight representations of the corresponding Lie(More)
Algebraic solutions of the sixth Painlevé equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked(More)
In 1991, one of the authors showed existence of quadratic transformations between Painlevé VI equations with the local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformation, up to fractional-linear transformations. The transformation is related to better known quadratic transformations(More)