Analytical research for transcendental numbers

这项研究的目的是用解析方法研究特殊函数，如超几何函数或多对数函数在代数点的值的无理性和超越度量等算术性质。关于这一点，我们可以邀请教授们进行讨论。G.Rhin，F.Beukers和L.Habsieger作为三年的海外合作者。对于高斯超几何函数，我们与M。哈特纳。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，它需要比非理性度量强得多的条件。我们可以确定一个实值函数成为无理型的充要条件。的确，存在着无数的实数，它们具有给定的无理性类型。此外，我们还可以确定…特定Fredholm型级数在特定有理点的值具有更多的无理性型。然而，关于(3/2)的分数部分的分布，我们没有得到任何新的结果，这使我们得以研究Ridout定理、Mahler Z数和Pisot数。这些课题需要继续研究。与本研究相关的是，上述研究者取得了以下成果。在一定的假设条件下，H.Saito证明了非齐次向量空间的Zeta函数的收敛性质和一个显式公式。M.Nagata研究了与Siegel的G-函数和G-算子有关的几个点上的Pade逼近，从而得到了G-函数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Riemann Zeta函数在奇整数处的特定值的Ramanujan公式之间的内在联系。M.Amou与K.Vaananen一起研究了一类多元函数方程解值的线性无关性。作为应用，它们在定量上改进了贝齐文的早期结果。较少

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