Toward a unified theory 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)，它们是用来处理经典的特殊函数，如Gauss超几何、Kummer合流超几何、Bessel、Hermite和Ary函数1：Ghf定义为Grassmanian Gr&lt；r，n&gt；r，n&gt；上的某些完整系统的解，并且它们具有形式意义上的积分表示。我们试图在De Rham理论的框架下理解这些积分，即某些上同调和同调群的上循环和圈的对偶。(1)对于广义Arey函数的完整系统表示为奇异轨迹外的可积完整联络的问题，我们在文[3]中计算了与…相关的有理扭曲de Rham复形的上同调群更多的是代表性。结果表明，除第r个上同调群和DIMH^r=_&lt；n-2&gt；C_&lt；r-1&gt；外，上同调群消失。此外，我们还提出了在Schur函数中给出一个H^r的基的猜想。(2)我们将广义Aary积分的积分域理解为P^r上具有被积函数所定义的支撑族的同调群的环。利用r维鞍点方法，我们在文[4]中证明了除第r个同调群外，其余同调群都是平凡的，并且第r个同调群在函数RANK_&lt；n-2&gt；C_&lt；r-1&gt；(3)当GHF由一维积分给出时(即Lauricella‘s F_D的合流情形)，我们证明了有理de Rham上同调群除H^1外都是平凡的，并明确地给出了H^1的基。2.已知其他合流型的特殊函数是由Gauss超几何函数通过称为合流的极限过程得到的。在文[5]中，我们证明了这种现象可以用李代数gl_n中的正则元集合中自然引入的分层层之间的邻接关系来解释，并且我们将上述极限过程推广到一般的GHF。较少

The objective of this project is to study the general hypergeometric functions (GHF) which were introduced by us to treat the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function.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. We try to understand these integrals in the framework of the de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here we treat this problem in the particular cases of GHF,the case of generalized Airy functions and the case of GHFs given by the one dimensional integrals.(1)In relation to the problem of expressing the holonomic system for the generalized Airy functions as the integrable holonomic connection outside of the singlar locus, we computed in [3] the cohomology group of the rational twisted de Rham complex associated with … More the representation. We showed that the cohomology groups vanish except for the r-th one and that dim H^r=_<n-2>C_<r-1>. Moreover we presented the conjecture that a basis of H^r is given in thems of the Schur functions.(2)We understand domains of integrations for the generalized Airy integrals as cycles of a homology group on P^r with the family of supports defined by the integrand. By using the r-dimensional saddle point method, we showed in [4] that the homology groups are trivial except for the r-th one and that r-th homology group forms a local system of Z-modules on the space of independent variables of the functions rank _<n-2>C_<r-1>.(3)In the case where the GHFs are given by the one dimensional integrals (in other terms, the confluent case of Lauricella's F_D), we showed that the rational de Rham cohomology groups are trival except for H^1, and gave a basis of H^1 explicitly.2 : It is known that the other special functions of confluent type are derived from the Gauss hypergeometric function by the limit processes called confluences. In [5] we showed that this phenomenon can be explained by the adjacency relations among the strata of the stratification naturally introduced in the set of regular elements in the Lie algebra gl_n. Furthermore we generalized the above limit process to GHF in general. Less