Integral representations of solutions of differential equations free from accessory parameters
type: Item
instance of: scholarly article
title: Integral representations of solutions of differential equations free from accessory parameters
zbMATH Open document ID: 1015.34081
author: Yoshishige Haraoka
published in: Advances in Mathematics
publication date: 2002-11-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34M35
zbMATH DE Number: 1836807
zbMATH Keywords: hypergeometric function
zbMATH Keywords: integral representation of Euler type
zbMATH Keywords: rigid local system
zbMATH Keywords: Euler-Darboux equation
zbMATH Keywords: Okubo normal form
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: aima.2001.2060
OpenAlex ID: W2007076268
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DOI: 10.1006/AIMA.2001.2060
description: scientific article; zbMATH DE number 1836807