Updated on : January 22, 2007
In order to make clear the meaning of confluences of the general
hypergeometric systems on the space of r by n matrices with
r
For each Painlev\'e system except the first one, we have a B\"acklund transformation group which is a lift of an affine Weyl group. In this paper, we show that the B\"acklund transformation groups for the 5th, 4th, 3rd, 2nd are successively obtained from that for the 6th by the well known degeneration or confluence processes.
It is shown that we can take coordinate systems determined by the B\"acklund transformations as ones of the manifolds of Painlev\'e systems and that the manifolds with parameters equivalent under the corresponding affine Weyl group are mutually isomorphic.
Certain degeneration processes among Painlev\'e equations are well known. In this paper, it is shown that these processes can be extended to those among the defining manifolds (or the spaces of initial condisions). Hamiltonian function on each chart preserves to be a polynomial in the process.
Elementary and somewhat geometric proof of Painle\'ve property for every Painlev\'e equation exept for the 1st one is given. We use the description of defining manifold for each Painlev\'e equation.
In this paper, certain symplectic descriptions of the defining manifolds for Painlev\'e equations from the 2nd to the 5th are given.
Some symplectic description of the defining manifold for the 6th Painlev\'e equation is given. Hamiltonian function on each chart is a polynomial. It is also shown that there exist no other algebraic Hamiltonian systems than the 6th Painlev\'e system on the manifold.