Toward a unified theory of special functions of several variables

本课题的目的是研究我们提出的一般超几何函数（GHF），对高斯超几何、Kummer超几何、Bessel、Hermite和Airy等经典的特殊函数有一个统一的认识，并对多变量的情况有一个自然的推广。1: ghf被定义为一类完整系统在Grassmannian Gr_<r, n>上的解，具有形式意义上的积分表示，其被积函数是P^r上的多值函数。为了得到关于GHF的显式结果，重要的是要在de Rham理论的框架中理解这种积分表示，即将其理解为某些上同调和同调群的环和环的对偶。这里，对于P^r上的积分，我们将同调群定义为局部有限同调群，并证明了对于某些子集P^r对，它与具有紧支持的相对同调群是同构的。利用这一结果，明确地计算了r=1时同调群的维数，并给出了同调群的一组基。2:对于具有规则奇点的GHF的最简单情况Beta函数B（alpha, Beta）和具有不规则奇点的GHF的最简单情况Gamma函数Gamma (alpha)，以下公式是众所周知的：B(alpha, Beta)B(-alpha, -beta)=2pii(<@ d71 (/)alpha@>D7+<@ d71 (/) Beta @>D7)(<@ d12pii (alpha+ Beta)@>D1-1(/)e<@ d12piialpha @>D1-1(e<@ d12piibeta @>D1-1)@>D7)；gamma(alpha)gamma(1-a)=<@D7pi(/)sinpialpha@> d7我们从de Rham理论的角度研究了理解上述公式的问题。明确地，我们试图将上述公式的右侧理解为上同交数与上同交数的乘积。对于由一维积分定义的GHF，通过选择上齐群的好基，显式地计算出其交矩阵。通过选择合适的基，我们可以证明交矩阵与一般超几何函数的变量无关。交数可计算的主要原因是，在de Rham复形的连接形式的每一个奇点上，良好的基具有类似于a型简单奇点的Jacobi环的平基的性质。少

The objecitve of this project is to study the general hypergeometric functions (GHF) which were introduced by us to give a unified understanding of the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function and to give a natural generalization to the case of several variables.1 : GHFs are defined as solutions of certain holonomic systems on the Grassmannian Gr_<r, n> and they have the integral representations in a formal sense whose integrand is a multivalued function on P^r. To obtain explicit resutis on GHF, it is important to understand this integral representation in the framework of de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here, for the integral on P^r, we defined the homology group as a locally finite homology group and then show that it is isomorphic to the relative homology group with compact supports for some pair of subsets P^r. Moreover … More , using this result, we computed explicitly, in the case r=1, the dimension of the homology group and gave a basis of the group.2 : For the Beta function B(alpha, beta), the simplest case of GHF with regular singularity, and for the Gamma function GAMMA(alpha), the simplest case of GHF with irregular singularity, the following formulas are well known :B(alpha, beta)B(-alpha, -beta)=2pii(<@D71(/)alpha@>D7+<@D71(/)beta@>D7)(<@D7-e<@D12pii(alpha+beta)@>D1-1(/)e<@D12piialpha@>D1-1(e<@D12piibeta@>D1-1)@>D7), gamma(alpha)gamma(1-a)=<@D7pi(/)sinpialpha@>D7We investigate the problem of understanding the above formulas from the viewpoint of de Rham theory. Explicitly we try to understand the right hand sides of the above formulas as a product of cohomological intersection number and the homological intersection number. For the GHF defined by the 1-dimensional integral, we computed explicitly the intersection matrix for the cohomoloy group by choosing its good basis.By the choice of good basis, we can show that the intersection matrix turns out to be independent of the variables of the general hypergeometric function. The main reason for the computability of the intersection numbers is that the good basis has, at each singular point of the connection form of the de Rham complex, the analogous properties to the flat basis of the Jacobi ring for the simple singlarity of A-type. Less