Confluence of cycles for hypergeometric functions on z2,n+1
Confluence of cycles for hypergeometric functions on z2,n+1
复制标题
z2,n 1 上超几何函数的循环汇合
DOI：
10.1090/s0002-9947-97-01471-2
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发表时间：
1997
影响因子：
1.3
通讯作者：
Y. Haraoka
中科院分区：
文献类型：
--
作者：
Y. Haraoka
一般型超几何函数是对经典合流超几何函数的推广，它允许由线性阿贝尔群的一个性质导出的积分表示。对于2 × （n + 1）矩阵空间上的超几何函数，通过一个极限过程构造其积分的环基，该极限过程称为合流过程。对周期矩阵的行列式进行了显式计算，以显示周期的独立性。在[KHT1]中介绍了λ型超几何函数。它包含了经典的合流超几何函数kummer、Bessel、Hermite和Airy函数及其在若干变量上的一般化作为专门化，有望成为专门化函数理论的重要研究对象。对于λ型超几何函数，我们得到了其核由线性阿贝尔群的特征给出的积分表示。然后用线性阿贝尔群描述了超几何函数的几个性质。然而，没有指定积分域（循环），我们只能研究形式性质。对于正则奇异情况，Aomoto [A2]， [A3]和Kita [Kt]研究了环，并建立了环的拓扑理论[IK1]， [IK2]。对于合流情况，没有对循环进行系统的研究。本文构造了具有一维积分表示的一般型超几何函数的环。在[KHT2]中，我们定义了控制超几何函数的线性阿贝尔群的合流，并由此得到了共环的合流。我们将在§2中说明，我们可以定义环的合流，使之与环的合流相容。然后，从基塔的正则奇异情况的扭环出发，逐步汇合，构造汇合情况的环（定理1.2.4）。我们的构造使得计算与超几何函数相关的周期矩阵的行列式成为可能。我们给出了定理3.1.3中行列式的显式形式，这表明了循环的独立性，编者于1994年7月28日和1995年1月9日以修订形式收到。1991数学学科分类。Primary 33C60, 33C65, 33C70。
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.