Study of special functions based on representation and invariant theories

研究的主要目的是寻找特殊函数世界背后的群体性理论背景，并利用特殊函数的语义。其中，“对偶”理论是我们研究的重点，它从表征理论和不变量理论的角度解释了许多现象。我们有了Capilli型恒等式，现在有了交流谐波振荡作为典型的研究，在这里和和都可以很好地运用网格原理。另一方面，对于罗恩和佩林的超越性，我们澄清了它们背后的群体操。这对深入研究这些函数有很大帮助。对于Capelli型恒等式，我们不仅得到了行列式，还得到了许多有趣的恒等式，包括永久恒等式和普富菲恒等式。此外，我们还发现了一些Capelli型恒等式对应于“群行列式”。不变的理论背景将这些恒等式与一些椭球函数联系起来。这是各种物体的统一。

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.