On Hypergeometric Functions and its Applications
Project/Area Number  10640163 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  Shiga University of Medical Science, Faculty of Medicine 
Principal Investigator 
TERADA Toshiaki University of Medical Science, Faculty of Medicine, Department of Mathematics, Professor, 医学部, 教授 (80025402)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2001 : ¥400,000 (Direct Cost : ¥400,000)
Fiscal Year 2000 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1999 : ¥400,000 (Direct Cost : ¥400,000)
Fiscal Year 1998 : ¥400,000 (Direct Cost : ¥400,000)

Keywords  Hypergeometric Function / Riemann Problem / Braid Group / Hypergeometric Representation / Complete Quadrilateral / 超幾何関数 / Riemannの問題 / ブレイド群 / 超幾何表現 / 完全四角形 / 組紐群 / 群の線型表現 / 保型関数 / 色付組紐群 / Burau表現 
Research Abstract 
(1) We presented some conditions under which the Wonskian of a finite sequence of functions does not vanish identically, and, by using them, we solved Riemann's problems for Larricella's F_D and Appell's F_4 without the nonvanishing of the Wonskian. They were solved by the author and respectively by Kato with stronger conditions about the orders of zeros at singular loci, which essentially assure the nonvanishing of the Wronskian. (2) At first, we intended also to prove the faithfulness of the hypergeometric representation of the braid group, but it have been proved not to be true. But we made a sketch of the proof of the faithfulness of that of the pure braid group. The procedure is as the following: the hypergeomtric representation of the pure braid group is the monodromy representation of the system of partial differential equations witch F_D satisfies and, if every parameters is rational, the solutions are periods of the algebraic curve υ^p = II^^<n+1>__<i=0>(ua_i)^<pi>. So the problem of the faithfulness is reduced to the problem : For a sequence of curves on a complex plain, if, every lift on every algebraic curve as above is 0homologuous, are they homotopically trivial under some conditions? At present, it is hard to declare it is solved, but it will be necessary only to polish up the details. And, using this, the conjugacy problem of the braid group is probably reduced to calculations of matrices, and there will be many contributions to the research of the hypergeometric functions.

Report
(6results)
Research Output
(7results)