The study of infinite product formulae for the Jackson integrals with Weyl group symmetry and their applications.

本研究的主题之一是研究无限乘积格的结构。为了得到无穷乘积表达式，我们需要两个事实。一个是关于参数的递推方程(二项关系)。另一类是当参数取无穷大时其渐近性态的主项。一旦我们有了递推关系，反复使用它，并利用它的渐近行为，我们就可以立即得到Jackson积分的无穷乘积表达式。但首先，为了实现这一点，我们需要递推关系本身和渐近行为的主项的显式形式。这两点是与根系有关的Jackson积分的难题。对于这两个点，直到我们发展了它们的计算方法，才进行了系统的研究。对于BCN型根系，我们最终发展了简单而基本的方法来获得它们：1.递推关系的两项分别对应于一些0次多项式和n次多项式。我们引入了一些中次i次的好的多项式，使得0&lt；i&lt；n.利用这些多项式对应的Jackson积分，我们得到了插值递推关系的方程。我们明确指出了上述步骤。由于Jackson积分的参数很多，参数取值的方向有很多选择。如果我们计算渐近行为，我们必须从它们中选择一个好的方向。在目前的研究中，我们找到了一个标准方向，使得人们可以用非常简单的方式计算渐近行为。

One of themes of the present research is to study the structure of an infinite product lattice. In order to obtain the infinite product expression we need two facts. One is the recurrence equation (two term relation) with respect to the parameters. The other is the principal term of its asymptotic behavior when we take the parameters to infinity. Once we have the recurrence relation, using it repeatedly and using the asymptotic behavior, we can immediately obtain the infinite product expression of the Jackson integral. But first of all, in order to carry this out, we need the recurrence relation itself and the explicit form of the principal term of asymptotic behavior. These two points are the difficult problem for the Jackson integral associated with the root systems. For these two points the systematical study had not been done yet until we developed the methods of calculating them.For the root system of type BCn, we eventually developed the simple and fundamental methods to obtain them as follows :1. The two terms of the recurrence relation are corresponding to some polynomials of degree 0 and degree n respectively. We introduced certain nice polynomials of middle degrees i such that 0<i<n. Using the Jackson integrals corresponding to these polynomials, we obtain the equations which interpolate the recurrence relation. We indicated the above procedure explicitly.2. Since the Jackson integral has many parameters, there are many choice of the direction for taking the parameters to infinity. We must choose a good direction from them if we calculate the asymptotic behavior. In the present research, we found a standard direction such that one can compute the asymptotic behavior in the very simple way.