Key words Rigid local system, system of Okubo normal form, monodromy representation MSC (2000) 34M35, 15A30For systems of differential equations of the form (xIn − T )dy/dx = Ay (systems of Okubo normal form), where A is an n × n constant matrix and T is an n × n constant diagonal matrix, two kinds of operations (extension and restriction) are defined. It is shown that every irreducible system of Okubo normal form of semi-simple type whose monodromy representation is rigid is obtained from a rank 1 system of Okubo normal form by a finite iteration of the operations. Moreover, an algorithm to calculate the generators of monodromy groups for rigid systems of Okubo normal form is given.

We give a method for constructing all rigid local systems of semi-simple type, which is different from the KatzDettweiler-Reiter algorithm. Our method follows from the construction of Fuchsian systems of differential equations with monodromy representations corresponding to such local systems, which give an explicit solution of the Riemann-Hilbert problem. Moreover, we show that every section of such local systems has an integral representation.

We study the structure of analytic continuation of solutions of an even rank system of linear ordinary differential equations of Okubo normal form (ONF). We develop an adjustment of the method by using the Euler integral for evaluating the connection formulas of the Gauss hypergeometric function 2F1(α, β, γ; x) to the system of ONF. We obtain recursive relations between connection coefficients for the system of ONF and ones for the underlying system of half rank.

No abstract

No abstract