Analytical research for transcendental numbers

本研究的目的是用解析的方法研究代数点上特殊函数（如超几何函数或多对数函数）值的无理性和超越测度等算术性质。关于这一点，我们可以邀请教授讨论。G. Rhin， F. Beukers和L. Habsieger在国外合作了三年。对于高斯超几何函数，我们可以显式地给出其对数导数的(n, n -1)- page近似。Huttner。数值应用将是下一个主题。另一方面，我们引入了“非理性类型”的概念，它需要比非理性测度更强的条件。我们可以确定实值函数成为无理数型的充分必要条件。确实存在具有给定无理数类型的不可数实数。此外，对于特定的Fredholm型序列在特定的有理点上的值，我们可以确定更多的非理性类型。但是我们没有得到关于（3/2）”的分数部分分布的新结果，这就引出了我们对Ridout定理、Mahler的z数和Pisot数的研究。这些课题应该继续研究。关于这项研究，上述调查人员得到了以下结果。H. Saito证明了预齐次向量空间的zeta函数在某些假设下的收敛性和一个显式公式。Nagata在与Siegel的g函数和g算子有关的几个点上研究了Pade近似，从而得到了g函数的有理值的一些密度估计。M. Katurada研究了某些q级数的渐近展开式与奇异整数处Riemann zeta函数特定值的Ramanujan公式之间的内在联系。M. Amou与K. Vaananen研究了若干变量泛函方程解值的线性无关性。作为一种应用，他们在定量上改进了先前的结果。少

The purpose of this research was to study arithmetical properties, for example irrationality and transcendental measures for the values of special functions, like hypergeometric or polylogarithmic functions, at algebraic points by analytic methods. Relating to this we could invite and discuss with Profs. G. Rhin, F. Beukers, and L. Habsieger as abroad collaborators during three years. Concerning Gaussian hypergeometric functions we could give explicitly (n, n -1)-Pade approximation for its logarithmic derivative, as a result of collaboration with Prof. M.. Huttner. The numerical application will be the next subject. On the other hand, we introduced a notion of 'irrationality type', which requires a much stronger condition than that of irratoinality measures. We could determine a necessary and sufficient condition that a real-valued function becomes an irrationality type. Indeed there exist uncountable real numbers which possess a given irrationality type. Moreover we could determine th … More e irrationality type for the values of specific Fredholm type series at specific rational points. However we could not get any new results about the distribution of he fractional part of (3/2)", which derived us to the study of Ridout's theorem, Mahler's Z-numbers, and Pisot numbers. These subjects should be studied continuously.Relating this research the above mentioned investigators have obtained the following results. H. Saito proved the convergence and an explicit formula, for the zeta functions of prehomogeneous vector spaces under some assumptions. M. Nagata studied Pade approximations at several points related to Siegel's G-functions and G-operators so that he obtained some density estimate on rational values of G-functions. M. Katurada studied intrinsic linkage between asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of Riemann zeta function at odd integers. M. Amou studied the linear independence of the values of solutions of certain functional equations in several variables with K. Vaananen. As an application they improved the earier result due to Bezivin quantitatively. Less