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02/06/01 Colloquium
PROFESSOR ALEXANDER V. KITAEV
DEPARTMENT OF MATHEMATICS
STEKLOV MATHEMATICAL INSTITUTE
DEPARTMENT OF MATHEMATICS
STEKLOV MATHEMATICAL INSTITUTE
Special Functions of the Isomonodromy Type, Riemann-Hilbert Problems, and Asymptotics
Abstract: Special functions of the isomonodromy type form a wide class of special functions which includes the Euler gamma function, Gauss hypergeometric function, and Painleve functions. The isomonodromy property of special functions allows one to obtain novel interesting results; in particular, higher order transformations for the Gauss hypergeometric function and algebraic solutions for the sixth Painlev? equation. Mathematical methods for study of the matrix Riemann-Hilbert conjugation problem in the complex plane have proved to be a powerful tool in the theory of so-called integrable systems, both continuous and discrete. Special functions of the isomonodromy type appear as model functions describing transition (caustic-type) phenomena for such systems. Two examples, namely, the nonlinear Schrodinger equation and Freud equation, in the theory of orthogonal polynomials, will be considered.