Yoshishige Haraoka

Canonical forms of differential equations free from accessory parameters

Systems of differential equations free from accessory parameters are defined and studied by Okubo [Seminar Reports of Tokyo Metropolitan University, 1987]. They are Fuchsian on the complex projective line, and there is an algorithm of determining monodromy representations for such systems. The Gauss hypergeometric equation, the generalized hypergeometric equation, the Pochhammer equation and a one-dimensional section of the Appell hypergeometric system

Monodromy representations of systems of differential equations free from accessory parameters

F_3

Confluence of cycles for hypergeometric functions on z2,n+1

are known to be reduced to such systems. Recently Yokoyama classified all the systems of differential equations which are irreducible and free from accessory parameters in terms of multiplicities of characteristic exponents. This paper presents canonical forms of all such systems and will define a new class of special functions.

Connection Problem for Regular Holonomic Systems in Several Variables

In a paper by Haraoka [SIAM J. Math. Anal., 25(1994), pp. 1203–1226], by following Okubo’s theory [K. Okubo, Seminar reports of Tokyo Metropolitan University, 1987], the canonical forms of all generic classes of Fuchsian systems of differential equations on

Three-dimensional representations of braid groups associated with some finite complex reflection groups

{\bf P}^1 ({\bf C})

Prolongability of ordinary differential equations

free from accessory parameters are obtained. Among them the explicit forms of six classes are new. In the present paper monodromy representations of the systems in the six classes are calculated. The technique employed is similar to one used to obtain the canonical forms. Hermitian forms invariant under those monodromy groups are also calculated. It turns out that the space of the invariant Hermitian forms for each system is real one-dimensional.

Middle convolution and deformation for Fuchsian systems

The hypergeometric function of general type, which is a generalization of the classical confluent hypergeometric functions, admits an integral representation derived from a character of a linear abelian group. For the hypergeometric function on the space of 2 × (n + 1) matrices, a basis of cycles for the integral is constructed by a limit process, which is called a process of confluence. The determinant of the period matrix is explicitly evaluated to show the independence of the cycles. Introduction The hypergeometric function of type λ is introduced in [KHT1]. It contains the classical confluent hypergeometric functions—Kummer, Bessel, Hermite and Airy functions—and their generalizations in several variables as specializations, and is expected to be a substantial object in the special function theory. For the hypergeometric function of type λ, we have an integral representation whose kernel is given by the character of a linear abelian group. Then several properties of the hypergeometric function are described in terms of linear abelian groups. However, without specifying domains of integration (cycles), we could study only formal properties. For the regular singular case, the cycles are studied by Aomoto [A2], [A3] and Kita [Kt], and there a topological theory is established for such cycles [IK1], [IK2]. For the confluent case, there is no systematic study of cycles. In this paper we construct cycles for the hypergeometric function of general type with 1-dimensional integral representation. In [KHT2] we have defined the confluence of linear abelian groups, which govern the hypergeometric functions, and then obtained the confluence of cocycles. We shall show in §2 that we can define the confluence of cycles so as to be compatible with the confluence of cocycles. Then by step by step confluence starting from the twisted cycles for the regular singular case owing to Kita, we construct cycles for the confluent case (Theorem 1.2.4). Our construction makes it possible to evaluate the determinant of the period matrix associated with the hypergeometric function. We give the explicit form of the determinant in Theorem 3.1.3, and this shows the independence of the cycles Received by the editors July 28, 1994 and, in revised form, January 9, 1995. 1991 Mathematics Subject Classification. Primary 33C60, 33C65, 33C70.

The generalized confluent hypergeometric function

We formulate the connection problem for regular holonomic systems in several variables on the basis of local monodromies. As examples, we solve the connection problem for Appell’s hypergeometric functions F1 and F2.

Integral Representations of Solutions of Differential Equations Free from Accessory Parameters

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in GL(3, ℂ). In many cases, we show the rigidity. For rigid representations, we give explicit forms of the representations, which turns out to be the monodromy representations of uniformization equations of Saito–Kato–Sekiguchi [Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math. 68 (2014) 181–221; On the uniformization of complements of discriminant loci, RIMS Kokyuroku 287 (1977) 117–137]. Invariant Hermitian forms are also studied.

Finite monodromy of Pochhammer equation

We extend the notion of deformation to inverse operations of restrictions of completely integrable systems to regular or singular locus, and call the extended notion prolongation. We show that a prolongability determines uniquely a Fuchsian ordinary differential equation of rank three with three regular singular points. This seems similar to that the deformation equation determines the accessory parameters as a function of the geometric moduli. Relations between prolongations and middle convolutions is also studied.