Kyoichi Takano

Research Interests by keywords

Recent (Pre)prints.

1. On Confluences of General Hypergeometric Systems, Tohoku Math. J., 58(2006), 1-31. coauthor: Kimura,H.

   In order to make clear the meaning of confluences of the general hypergeometric systems on the space of r by n matrices with r<n, we generalize our previous definition of hypergeometric systems. We define the systems for the centralizers of regular elements of the linear Lie algebra of order n and their weights. Then we give explicitly confluence processes of among regular elements, which yield automatically those among hypergeometric systems. In the last section, examples for one-variable cases and two-variable cases are given.

2. Hierarchy of Bäcklund Transformation Groups of the Painlevé Systems, J. Math. Soc. Japan, 56(2004), 1221-1232. coauthors: Suzuki, M., and Tahara, N.

   For each Painlevé system except the first one, we have a Bäcklund transformation group which is a lift of an affine Weyl group. In this paper, we show that the Bäcklund transformation groups for the 5th, 4th, 3rd, 2nd are successively obtained from that for the 6th by the well known degeneration or confluence processes.

3. Bäcklund Transformations and the Manifolds of Painlevé systems, Funkcial. Ekvac., 45(2002), 273-258. coauthors: Noumi, M., and Yamada,Y.

   It is shown that we can take coordinate systems determined by the Bäcklund transformations as ones of the manifolds of Painlevé systems and that the manifolds with parameters equivalent under the corresponding affine Weyl group are mutually isomorphic.

4. Confluences of Defining Manifolds of Painlevé Systems, Tohoku Math. J., 53(2001), 319-335.

   Certain degeneration processes among Painlevé 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.

5. Defining Manifolds for Painlevé Equations, "Toward the exact WKB analysis of differential equations, linear or non-linear" (Eds. C.J. Howls, T. Kawai, and Y. Takei), 261-269, Kyoto Univ. Press, Kyoto, 2000.

   Elementary and somewhat geometric proof of Painlevé property for every Painlevé equation exept for the 1st one is given. We use the description of defining manifold for each Painlevé equation.

6. On Some Hamiltonian Structures of Painlevé Systems, II, J. Math. Soc. Japan, 51(1999), 843-866, coauthors: T. Matano, A. Matumiya

   In this paper, certain symplectic descriptions of the defining manifolds for Painlevé equations from the 2nd to the 5th are given.

7. On Some Hamiltonian Structures of Painlevé Systems, I, Funkcial. Ekvac., 40(1997), 271-291, coauthor: T. Shioda

   Some symplectic description of the defining manifold for the 6th Painlevé 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 Painlevé system on the manifold.