Open Collections

Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral

Banff International Research Station for Mathematical Innovation and Discovery

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions. This talk is based on a joint work with Toshiyuki Mano (Math. Z. 2016).

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: Hitotsubashi University

2017-04-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61091

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/