Study of special functions based on representation and invariant theories

研究的主要目的是找出特殊功能世界背后的群体理论背景，并利用特殊功能的协调性。其中“对偶”理论是我们研究的重点，它从表象理论和不变量理论的角度解释了许多现象。我们有Capilli类型的恒等式，现在交换的Harnics振荡作为典型的调查，其中And Pain非常好地工作于网格原理。另一方面，对于Rons和Pain Lene Transcendents中的性交亢进症，我们已经解释了他们背后的普通体操。对于Capelli型恒等式，我们得到了许多有趣的形式，包括永久恒等式和Pfuffian恒等式，不仅对于行列式，而且我们还找到了对应于“群行列式”的Capelli型恒等式。不变的理论背景将这些恒等式归结为一些蝶形函数。各种物体都有某种程度的统一性。

The main object of the research is to find the group theoretical background behind the world of special functions and to utilize the symonetious for the special functions. Among them the theory of "dual pairs" is the key to our study, which explains many phenomina from the view-point of representation theory and the theory of invariants. We have Capilli type identities, now commtative harnomic oscillatws as the typical investizations where and pains work very well as the griding principle. On the other hand, for the hyprogeinctic from Rons and Pain lene transcendents, we have claified the grop gymmetric behind them. The helps a lot for deeper investigations of these fnctions.As for the Capelli type identities, we got many interesting formlas including permanets and Pfuffians, not only for the determinants, Furthermore we found some Capelli type identity corresponding to the "group determinant". The invariant theoretic backgroud conneits these identities to some sphenicel functions. There are sort of unification of various objects.