On Hypergeometric Functions and its Applications

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(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.