Toward a unified theory of special functions of several variables

本项目的目的是研究我们引入的一般超几何函数(GHF)，以统一理解经典的特殊函数，如Gauss超几何、Kummer合流超几何、Bessel、Hermite和Ary函数，并给出多元情形的自然推广。1：Ghf定义为Grassmanian Gr&lt；r，n&GT上某些完整系统的解；并且它们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。要得到GHF上的显式结果，重要的是要在De Rham理论的框架下理解这种积分表示，即某些上同调和同调群的上圈和圈的对偶。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，并证明了它与某些子集对P^r具有紧支集的相对同调群同构.此外，…此外，利用这一结果，我们在r=1的情况下，显式地计算了同调群的维度，并给出了群的基2：对于具有规则奇异性的GHF的最简单情况--Beta函数B(α，β)，以及对于具有不规则奇性的GHF的最简单的情况--伽玛函数GAMMA(α)，众所周知的公式如下：B(α，β)B(-α，-beta)=2pii(&lt；@D71(/)alpha@&gt；D7+&lt；@D71(/)beta@&gt；D7)(&lt；@D7-e&lt；@D12pii(alpha+beta)@&gt；D1-1(/)e&lt；@D12piialpha@&gt；D1-1(e&lt；@D12piibeta@&gt；D1-1)@&gt；D7)，gamma(alpha)gamma(1-a)=&lt；@D7pi(/)sinpialpha@&gt；D7We从德罗姆理论的角度研究了理解上述公式的问题。显式地，我们试图将上述公式的右侧理解为上同调交数和上同调交数的乘积。对于由一维积分定义的GHF，我们通过选择上同调群的良基来显式计算上同调群的交矩阵，通过选择良基，我们可以证明交矩阵与一般超几何函数的变量无关。交数的可计算性的主要原因是良基在De Rham复形的联结形式的每个奇点上具有类似于雅可比环的平坦基的性质，用于A型的简单奇异性。较少

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