Integral representations of solutions of differential equations free from accessory parameters
An Okubo normal form is a linear system of ordinary differential equations of the form \((*)\) \((xI_n-T)(du/dx)=Au\) where \(T\) and \(A\) are constant \(n\times n\)-matrices (\(T\) is diagonal), and \(u\) is an \(n\)-vector of unknown functions. The author reproves first Okubo's observation that every Fuchsian differential equation on \({\mathbb C}P^1\) can be obtained as a subsystem of a system \((*)\). Fuchsian differential equations free from accessory parameters correspond to rigid local systems by taking their monodromy representations. The author describes four Yokoyama operations on Okubo normal forms; they change the rank \(n\) of the system. He shows that every system \((*)\) which is irreducible and free from accessory parameters is obtained from a rank \(1\) system \((*)\) by a finite iteration of these operations. (Rank \(1\) systems can be explicitly integrated.) The proof being constructive, it shows how one can find integral representations of the solutions of Okubo normal forms free from accessory parameters. The author establishes relations among a system in Okubo normal form, the Euler-Darboux equation and Yokoyama's Pfaffian system.
scientific article; zbMATH DE number 1836807
scientific article