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

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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.