Integral Representations of Solutions of Differential Equations Free from Accessory Parameters

Abstract

We show that every accessory parameter free system of differential equations of Okubo normal form has integral representation of solutions. The proof is constructive; we study the change of solutions under the operations}the extension and the restriction, which have been introduced by Yokoyama [Construction of systems of differential equations of Okubo normal form with rigid monodromy, preprint] in order to construct every such system of differential equations. Several examples are given. # 2002 Elsevier Science (USA)

Construction of rigid local systems and integral representations of their sections

“…Our method follows from the construction of Fuchsian systems of differential equations with monodromy representations corresponding to rigid local systems of semi-simple type, which give an explicit solution of the Riemann-Hilbert problem for such representations (Proposition 1.2 and Theorem 1.6). Moreover by applying the result in [6] we show that such Fuchsian systems of differential equations have integral representations of the solutions (Theorem 3.2). In terms of local systems, this result means that every section of every rigid local system of semi-simple type has an integral representation.…”

“…In [6] we have shown that every irreducible rigid system of ONF of semi-simple type has an integral representation of the solutions of Euler type. Suppose that the system (1.3) is irreducible, rigid and of semi-simple type.…”

“…, p}. The restriction operation R of the system (1.3) is defined by the system 6) whereŤ andB are (n − n k ) × (n − n k )-matrices obtained from T and B, respectively, by removing the k-th block row and the k-th block column. It turns out that the system (1.6) is a system of ONF of semi-simple type of dimension n − n k which has the spectral type…”

“…In particular we have given an algorithm for obtaining all such systems of ONF in [16], and have traced the change of solutions of the systems in the process of the algorithm in [6].…”

“…On the other hand we have established the theory of systems of ONF with rigid local systems of semi-simple type as monodromy representations ( [2]- [6], [11], [12], [14]- [16]). In particular we have given an algorithm for obtaining all such systems of ONF in [16], and have traced the change of solutions of the systems in the process of the algorithm in [6].…”