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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.

Confluence of cycles for hypergeometric functions on 𝑍_{2,𝑛+1}