On Hypergeometric Functions and its Applications

(1)给出了有限函数序列的Wonskian不完全消失的几个条件，并利用这些条件，在Wonskian不消失的情况下，解决了Larricella的F_D和Appell的F_4的Riemann问题。这两个问题分别由作者和加藤用更强的关于奇异轨迹处零阶的条件解出，从本质上保证了朗斯基行列式的不灭性。(2)起初，我们也想证明辫群的超几何表示的忠实性，但它已被证明是不正确的。但我们对纯辫群的可靠性证明做了一个草图。其过程如下：纯辫群的超几何表示是F_D满足的偏微分方程组的一元表示，如果每个参数都是有理的，其解是代数曲线υ^p = II^^<n+1>__<i=0>(u-a_i)^<pi>的周期。因此，忠实度的问题被简化为：对于复平面上的曲线序列，如果上述每条代数曲线上的每一个升力都是0-同调的，那么它们在某些条件下是否同伦平凡？目前，很难宣布问题已经解决，但只需要对细节进行润色。利用这一方法，将编织群的共轭问题简化为矩阵的计算，对超几何函数的研究有很大的贡献。

(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 non-vanishing 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 non-vanishing 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>(u-a_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 0-homologuous, 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.