  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 for functions 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.
